The model also needs to specify the temporal dynamics of the environment and the relation between S and E. The simplest assumption is that successive environmental states are uncorrelated, and occur with probability p(E) and that signals are derived from a conditional probability p(S|E), as illustrated in Figure 13.1B where p(S|E) = δ(S|E) is equivalent to S ≡ E. This is sufficient to demonstrate hedging or discuss the value of signaling and in the examples below we therefore make this simplifying assumption by default. More generally, to address issues of inheritance where R<sub>t</sub> depend on R<sub>t-1</sub>, we may assume a discrete-time Markov process where the state of the next environment depends only of the previous one, with transition probabilities p(E<sub>t</sub>|E<sub>t-1</sub>) where E<sub>t</sub> denotes the state of the environment at time t = 1, 2, . . . . Even more generally, we may also want to account for the feedback that the population exerts onto its environment and consider that E<sub>t</sub> depends on the size and composition of the population.

Finally, we need to specify the dynamics of the population itself. Between time points t and t + 1, a cell adopting phenotype R in the context of environment E<sub>t</sub> either dies or survives and may additionally produce offsprings. This is summarized by a quantity f(R, E<sub>t</sub>) ≥ 0 that indicates the mean number of descendants at time t + 1 of an individual with phenotype R in environment E<sub>t</sub> (possibly including the individual itself). Given that u(R|S<sub>t</sub>) denotes the fraction of cells or probability of the organism adopting phenotype R based on sensed state S<sub>t</sub>, a population is therefore expected to globally increase (or decrease) in size by a factor

$$
A_t = \sum_R f(R, E_t)u(R|S_t) \qquad (13.1)
$$

that depends both on the strategy u and the current environmental state E<sub>t</sub>. This factor A<sub>t</sub> is a stochastic variable as it depends on the stochastic variables E<sub>t</sub> and S<sub>t</sub>. More explicitly, if N<sub>t</sub> denotes the size of the population at time t, this size will increase or decrease to N<sub>t+1</sub> = A<sub>t</sub>N<sub>t</sub> at time t + 1 (in average). We can in this way account for the dynamics of population growth and then ask what is an "optimal" strategy u(R|S) that leads to, for example, the largest population size over a given time interval. Compared to the deterministic setting, however, this is not yet a well-formulated problem as the population size varies with time and therefore generally depends on the particular sequence of environments E<sub>0</sub>, . . . , E<sub>t</sub>, which is in turn stochastic. Thus, we need to extend the concept of optimality to the stochastic regime. We examine this question in the next section.

13.4 Optimization in uncertain environments

In the previous section, we used notation A_t to denote the fractional increase or decrease in population size given that strategy u(R|S_t) is adopted in environment E_t. An alternate name for this quantity is the instantaneous growth rate. It follows from recursion that, given an initial population size of N_0 at time t = 0, the population size at time t is given by

$$
N_t = A_t A_{t-1} \cdots A_1 N_0 \tag{13.2}
$$

where A_t depends on the environmental state E_t and is therefore a stochastic variable when the environ-
ment varies stochastically. Here the choice of an objective criterion is fundamentally linked to the time
scale at which growth is considered.

13.4.1 Long-term versus short-term optimization

At the shortest time scale, maximization of population growth over a single time step corresponds to adopting the distribution u(R|S<sub>t</sub>) that maximizes the arithmetic mean E[A], where A denotes the random variable whose realization at time t is A<sub>t</sub> (Box 1). This maximum is typically achieved by a population where all individuals adopt the same optimal phenotype – the phenotype R maximizing E[f(R, E<sub>t</sub>)u(R|S<sub>t</sub>)] = Σ<sub>E<sub>t,S</sub></sub> P(S|E)P(E)f(R, E)u(R|S). In the example of persistent cells, this strategy would correspond to having all cells in a growing state if the most likely environment is an absence of antibiotics. This strategy is extremely risky if these growing cells cannot survive an episode of antibiotics, which would therefore lead to extinction of the population. Taking into account the rare but important events of high antibiotics concentration requires taking a long-term perspective. Remarkably, in the long-term the problem becomes effectively deterministic due to the law of large numbers. The best known example of a law of
<figure><img src="image_92.png" /><figcaption>Figure 14.5: Trade-off between elastic energy stored in the tissue and viscous energy dissipated in the air circulation (exercise regime, computed from our model).</figcaption></figure>

### 14.2.3. The energy expenditure or the cost of breathing

Breathing is part of the basal metabolism, meaning that it is a regular and mandatory energy cost for the maintenance of the body. Yet, natural selection, one of the main processes driving evolution, tends to select for minimal energetic cost so that the organisms can allocate most of their resources to their reproduction [579]. Hence, in order to understand breathing, it is important to determine the origin of the energetic costs and how they are affected by the breathing process. We already pointed out that diffusion, considered from the lung point of view, is a passive process. So, most of the energetic costs involved in the lung function arise from the process of ventilation. Energy is spent through the action of the muscles on the lung. This action has two main effects: it deforms the tissues and it displaces the air along the bronchial tree.

On the one hand, the tissues are deformed due to the action of the thoracic muscles, especially the diaphragm. This deformation is considered as elastic in the normal range of ventilation [580], and energy is dissipated along the displacement of the tissues. On the other hand, as every gas, air acts as a fluid with specific viscosity. As the bronchial tree is an assembly of a high number of narrow tubes with decreasing size, the energy spent for the displacement of the air in the bronchial tree is dominated by the energy dissipated by the friction of air in the bronchi. The air kinetic energy is negligible relatively to the dissipation. This can be summarized in term of the power spent by the muscles (energy per unit of time):

$$ \underbrace{\mathcal{P}_m}_{\text{muscle power}} \approx \underbrace{\mathcal{P}_e}_{\text{elastic power}} + \underbrace{\mathcal{P}_a}_{\text{air viscous dissipation}} . $$

These quantities depend on several lung characteristics, on the breathing frequency *f* and on the amount of air inhaled during on breath cycle *V*<sub>T</sub>. This raises the trade-off shown in Figure 14.5 and, using optimization theory, optimal ventilation frequencies and tidal volume can be be predicted.

The viscous dissipation of air in the bronchial tree is characterized by the lung hydrodynamic resistance *R*, which is directly related to the geometry, size, number and structuring of the bronchi [580]. The hydrodynamic resistance is a physical quantity that represents how the energy put in the system is divided between kinetic energy and heat energy. It connects the volume of air displaced per unit of time, also called air flow *F*, to the force per unit of surface applied to the air, also called air pressure *p*<sub>a</sub>: *p* = *R* *F*. For the same pressure applied on the lung, the higher the hydrodynamic resistance, the lower the air flow and the higher the dissipation. Then, the power dissipated by viscous friction of the air
Σ Math box 13.A Arithmetic versus geometric mean and logarithmic utility

Additive random processes are governed by the law of large numbers: the sum of many random variables scales with their arithmetic mean. In finance and biology, returns are compounded and growth is a multiplicative process. This is fundamentally different: the typical outcome is no longer described by the arithmetic mean but by the geometric mean [410]. A simple example illustrates this difference. Imagine a succession of environments in which the population either doubles or is reduced by 2/3, with same probability. This corresponds formally to a population size increasing as $N_t = A_t \dots A_1 N_0$ where $A_t = 2$ (doubling) with probability 1/2 and $A_t = 1/3$ (2/3 dying rate) with probability 1/2. The arithmetic mean is 7/6 which is > 1 and suggests that the population will grow. But as each outcome has the same probability, the typical growth over t generation is actually given by $2^{t/2}(1/3)^{t/2} = e^{t\Lambda}$ with $\Lambda = (1/2) \ln(2/3)$ which is < 0: the population will in fact most likely go extinct. Mathematically, taking the log turns the product into a sum to which the central limit theorem applies. More intuitively, the arithmetic mean is dominated by very rare events. Historically, the importance of the geometric mean for estimating risk was first understood by Daniel Bernoulli in the context of games [411, 412]. Later, it has been the subject of many debates in finance [411], reflecting the fact that alternative utility functions over which to optimize may be more appropriate when considering a short temporal horizon or when accounting for different degrees of risks.

large number applies to the sum $A_1 + \dots + A_t$ of t random variables $A_i$, which almost certainly behaves as $t\mathbb{E}[A]$ as $t \to \infty$. Here, the problem involves a product of random variables and a similar but different law of large number applies: the product $A_1 \times \dots \times A_t$ does not typically behave as $(\mathbb{E}[A])^t$ but instead as $\exp(t\mathbb{E}[\ln A])$ where $\mathbb{E}[\ln A]$ is known as the (Box 1). This corresponds to the intuition that population size typically grows exponentially in the long run, $N_t \sim e^{\Lambda t} N_0$, with a well-defined long-term growth rate

$$ \Lambda = \mathbb{E}[\ln A] = \sum_{E,S} p(S|E)p(E) \ln \left( \sum_{R} f(R, E)u(R|S) \right), \quad (13.3) $$

that is predictable despite the stochasticity of the environment.

Biologically, therefore, maximizing the geometric mean is equivalent to maximizing the long-term growth rate of the population. This is the relevant measure of fitness in the long-term from an evolutionary point of view, because of two populations with growth rates $\Lambda_1$ and $\Lambda_2$, the one with $\Lambda_1 > \Lambda_2$ will almost certainly exponentially outnumber the other.

The simple example of persistence that we introduced previously illustrates well how maximizing the long-term is different from optimizing the instantaneous growth rate. The arithmetic mean $\mathbb{E}[A]$ is indeed maximized by $u_d = 0$ when $p_a < 1/2$, which leads to certain extinction unless $p_a = 0$. This remains true for general models including multiple environmental states and sensing that conveys information about the environment through conditional probability $p(S|E)$. Using the long-term growth rate $\Lambda$ as a measure of fitness, it is then possible to quantify the value of information $S$ by comparing the optimal growth rate that can be achieved in presence of $S$ to that in its absence. Remarkably, for special limits of the model, corresponding to Kelly's horse-race model (Box 2), this value is given by some of the same quantities that appear in Shannon's theory of communication (Box 3).

### 13.4.2 Trade-offs at intermediate time scales

So far we considered two extreme limits of immediate and infinite time scales under one important assumption: the population is always large enough to escape extinction. Eq. (13.2) is indeed valid only for large $N_t$ and does not apply anymore when $N_t \sim 1$, in which case the population size is subject to stochastic effects, called demographic noise in population biology. In our analogy with finance, the eventuality of $N_t = 0$ with no possible recovery corresponds to a risk of bankruptcy.

When considering long time scales, a population with $\Lambda > 0$ will either become extinct or grow exponentially. In this later case, demographic noise is eventually negligible and our approach valid. At intermediate time scales, however, population sizes $N_t$ may deviate substantially from $N_0e^{\Lambda t}$ predicted by exponential growth, and may become extinct ($N_t = 0$) as a result. To quantify these deviations, note that for the model
inside all the bronchi can be estimated by $P_a = pF = RF^2$. By assuming in our case that the velocity of the air follows a sinus function, we can deduce the power dissipated by viscous friction as follows:

$$P_a = \frac{1}{4} R (\pi f_b V_T)^2 .$$

<figure><img src="image_93.png" /><figcaption>Figure 14.6: Total power expenditure during ventilation (W) as a function of the respiratory frequency (min⁻¹) for different intensities of exercise. Dots correspond to the optimal ventilation frequency i.e., that minimizes the dissipated power. Adapted from [581].</figcaption></figure>

The elastic power is characterized by the compliance $C$ of the lung, that relates the force per unit of surface applied by the muscles ($p_m$) to the volume change of the lung [582]. The compliance depends on the lung's volume, especially when the deformation of the lung is high although the compliance can be considered constant while healthy. That is why, in our case, we assume that the compliance is a constant and we neglect the non-linearities arising at large lung's deformations [583]. The elastic power can be estimated by integration of the volume along the inspiration phase and it gives us,

$$P_e = \frac{V_T^2 f_b}{2C}.$$

Finally, the total energetic cost of breathing $P$ can be written as the sum of the power dissipated by viscous friction $P_a$ and the elastic power $P_e$. The total power has to be minimized relatively to the tidal volume $V_T$ and the breathing frequency $f_b$ with a constraint on the oxygen flow to blood that has to match the oxygen flow demand (see Equation 14.1). Thanks to our model previously defined, we can compute the oxygen flow to blood as a function of tidal volume and breathing frequency and compare it to the oxygen flow $\dot{V}_{O_2}$ requested by the body at the regime considered.

Our model predicts (see Figure 14.6), for a human at rest, an optimal breathing frequency of 12.2 breaths per minute and an optimal tidal volume of 497 mL, which are very close to the average physiological values [574]. The model exhibits a robustness in term of frequency perturbation around the optimal. A 5% shift in the energy brings the frequency into a range between 8 breaths per minute up to 18.5 breaths per minute. This effect is due to the fact that, at low regimes, a low tidal volume $V_T$ is sufficient to perform an optimal ventilation. When the exercise intensity increases, the power profiles as a function of the frequency become steeper and steeper and focus the optimal value within a tighter region. It implies that a shift from the optimal configuration at high intensities is predicted to be costly in term of energy spent. This behavior is fully compatible with the fact that the control of ventilation is stronger at exercise, preventing even
Σ Math box 13.B Kelly's optimal betting strategy

In 1956, [413] extended the work of on communication to the field of. This classic model has important implications for investment strategies in and beyond. In the context of biology, Kelly's paper led to a clarification of the notion of value of information which is described in Box 13.C

Let us recall the basic elements of Kelly's horse race. The odds paid by the bookmaker when the horse $x$ wins is $o(x)$, and the probability for this to happen is $p(x)$. A gambler can distribute his/her bets on the different horses, and $b(x)$ is the fraction of the bet set on horse $x$. Thus, a *strategy* of the gambler is defined by a vector of bets $b$ of $M$ components $b(x)$. At every race, the gambler invests his/her entire capital on all horses, so that $\sum_{x=1}^{M} b(x) = 1$, always betting a non-zero amount on all horses. Since no bet is zero, there is a well-defined vector of the inverse of the odds paid by the bookmaker denoted $r$. When the odds are *fair*, the bookmaker does not keep any of the invested capital and as a result $\sum_{x=1}^{M} r(x) = 1$.

At each time $t$, one horse, which we call $x$, wins with probability $p(x)$. As a result, the capital at time $t+1$ is updated according to $C_{t+1} = \frac{b_x}{r_x} C_t$. As explained previously, this multiplicative process is best studied by considering instead the log of the capital, $\log-cap(t) \equiv \ln C_t$, which satisfies the assumptions of the law of large numbers when races are independent. In these conditions, $\log-cap(t) \equiv \ln C_t$ converges on long times towards the growth rate $W(\mathbf{b}, \mathbf{p})$ where

$$W(\mathbf{b}, \mathbf{p}) = \sum_{x} p(x) \ln o(x)b(x). \qquad (13.4)$$

This growth rate can be rewritten using an information theoretic measure between two probability distributions, $\mathbf{p}$ and $\mathbf{q}$, called the Kullback-Leibler divergence and defined by

$$D_{KL}(\mathbf{p}, \mathbf{q}) = \sum_{x} p(x) \ln \frac{p(x)}{q(x)}. \qquad (13.5)$$

One can show that this quantity is a non-negative measure between the two probability distributions. With this notation, the growth rate can be rewritten as

$$W(\mathbf{b}, \mathbf{p}) = D_{KL}(\mathbf{p}\|\mathbf{r}) - D_{KL}(\mathbf{p}\|\mathbf{b}), \qquad (13.6)$$

It follows from this equation that the strategy $\mathbf{b}^* = \mathbf{p}$ is optimal. This strategy, known as Kelly's strategy or proportional betting, overtakes any other strategy in the long-term as illustrated in Fig. 13.2.

This formulation shows that the growth rate is the difference between the distance of the bookie's estimate from the true distribution and the distance of the gambler's estimate from the true distribution. Hence, the gambler makes money if they have a better knowledge of the winning probabilities than the bookie. The optimal long term growth rate is the positive quantity :

$$W^*(\mathbf{b}, \mathbf{p}) = D_{KL}(\mathbf{b}\|\mathbf{r}). \qquad (13.7)$$

Kelly's horse race model is formally a particular case of the model introduced in the main text when considering that one, and only one phenotype $R = R(E)$ can grow in any given environment $E$, such that $f(R, E) = f(E)$ if $R = R(E)$ and $0$ otherwise. Horses $x$ may then be interpreted as both the environments $E$ and their associated phenotypes $R(E)$ so that $u(R) = b(x)$ and $f(E) = o(x)$. In biology, but also in finance where $R$ is interpreted as an asset, there is generally no one-to-one correspondence between environments $E$ and phenotypes $R$ and multiple phenotypes (assets) may grow (have non-zero return) in any given environment. The optimal strategy is then no longer necessarily proportional betting as illustrated in the example of persistence presented in the main text and as also shown in Ref. [414].

defined in the main text where there are no correlations of the instantaneous growth rate $A_t$, the central limit theorem imposes that the quantity

$$\Delta_t = \frac{1}{\sigma\sqrt{t}} \left( \ln \frac{N_t}{N_0} - t\Lambda \right), \qquad (13.8)$$

converges on long times towards a Gaussian distribution of unit variance, where $\sigma$ is the standard deviation of the instantaneous growth rate. It follows from this property that

$$\sigma^2 = \frac{1}{t} \mathrm{Var} \left( \ln \frac{N_t}{N_0} \right), \qquad (13.9)$$
<figure><img src="image_94.png" /><figcaption>Figure 14.7: Distribution of total sleep duration (h) in mammals, based on data from Savage & West [585]. The data are best fitted by the curve indicated in light gray.</figcaption></figure>

talking.

The question of the optimal conditions of ventilation in human leads naturally to a series of extensions that need to be considered. We have seen previously that the optimization under constraints occurs in almost every organ in all the living beings. Thus, could we expect the present model to be extended to all mammals, as the control of ventilation is, more than probably, present in the whole mammalian class?

## 14.3. Allometric scaling laws for respiration and ventilation

The answer to this question of generalization leads us to a vast scientific question that will bring us back to the late 19th century and which is still open on many aspects.

### 14.3.1. The emergence of scaling relations in nature

In 2007, Savage and West published a seminal work in which they present a collection of data of sleep duration in a set of mammalian species. Among other major results, their analysis confirmed the previous observation [584] that the larger the animal, the shorter the duration of its sleep cycle [585]. More precisely, the sleep duration correlates negatively with the body mass of the mammal and follows, based on the data from Savage & West, an interesting exponential law of the form $t_s = 10.1 M^{-0.103}$, with $t_s$ the sleep duration in hours during a 24 hours period and M the body mass of the mammal in kilograms, as seen in Figure 14.7 [585].

Thus, by taking the *log* of both sides of the equation, one can write this sleep-to-mass relation as $\log t_s = \log 10.1 - 0.103 \log M$ i.e., a linear relation between the logarithm of the sleep duration and the logarithm of the mass of the animal, see Figure 14.8. As we will see later, this type of exponential relation is now referred, in ecological sciences, as an *allometric* scaling. In general, an allometric law will write $Y = Y_0 M^b$, where Y is the studied – physiological, morphometric – property, M the mass of the living organism, $Y_0$ and b the allometric prefactor and exponent, respectively [586, 587].

Actually, the concept presented by Savage & West is far from being recent. The history of the study of allometric relations dates back to the 19th century. Scientists from various disciplines started to analyze the changes in shape and form of living beings in relation with their overall size [588]. Let us read the fascinating story told by Jean Gayon about the origins of the allometric concept [589].
<figure><img src="image_120.png" /></figure>

<figure><img src="image_121.png" /><figcaption>Figure 13.2: Kelly's optimal strategies – Evolution of the log-capital (A) or of the capital itself (B) as a function of the number of races for Kelly's optimal strategies (red curve) and for a non-optimal strategy (yellow curve). In log-scale, the straight lines have the slope of the corresponding growth rate for each strategy. Note that the fluctuations in Kelly's strategy can in fact be quite large, when plotted in normal scale instead of log-scale.</figcaption></figure>

measures the deviation from exponential growth. This quantity is therefore a natural measure of risk, known in finance under the name of volatility. To understand at which time scale this risk is important, we consider Eq. 13.8, assuming $\Delta_t$ is of the order one. Risk will be important, when the term associated with fluctuations, which is of the order of $\sigma\sqrt{t}$ will be larger than the term associated with exponential growth, which is $t\Lambda$. This will happen when $t \ll (\sigma/\Lambda)^2$: the risk is relevant at intermediate time scales, long-enough for the central limit theorem to apply but not too long for deviations from exponential growth to become negligible.

This measure of risk has well known drawbacks in finance: it is symmetrical with respect to losses and gains, which does not conform to the intuitive notion of risk, and furthermore typical fluctuations are often non-Gaussian. Nevertheless, the volatility is still an important notion in the study of optimization of portfolios [418]. In this context, Markowitz introduced plots of the volatility $\sigma$ as a function of the mean growth rate, which define the so-called "efficient frontier". This representation illustrates graphically a fundamental trade-off that exists between the maximization of the mean return and the minimization of the variance (or risk). The point of zero volatility is a risk-free strategy, which corresponds to dormant states in biology.

This trade-off is naturally present in Kelly's model introduced in Box 2. Indeed, Kelly's strategy is based on the maximization of the long-term growth rate, but at intermediate times the capital can deviate significantly from the expected exponential growth as shown in Figure 13.2. Prominent economists, such as Samuelson, strongly opposed the use of Kelly's criterion in finance precisely for that reason [419]. In practice, however, investors can mitigate this risk by using Kelly's criterion for only a fraction of the bets [420]. The resulting strategy has reduced fluctuations, and at the same time, a reduced growth rate. Another consequence of the trade-off is that the risk near the optimal strategy (Kelly's strategy) can be reduced significantly provided one is ready to sacrifice a small amount of growth rate, an important lesson for gamblers and investors. In order to build systematically improved gambling strategies with a reasonable amount of risk in Kelly's model, one can introduce an objective function that is a linear combination of the growth rate with the volatility of Kelly' model, $\sigma_W$, weighted by a risk aversion parameter $\alpha$ [421]. The method is illustrated in Problem 13.4 for the two-horse version of Kelly's model. By optimizing this objective function, one builds the Pareto diagram shown in Fig. 13.3 when varying the parameter $\alpha$.
<figure><img src="image_95.png" /><figcaption>Figure 14.8: Distribution of total sleep duration (h) in mammals in log-log plot, based on data from Savage & West [585]. The data are best fitted by the line indicated in bold black. Confidence intervals are presented as light gray zones along the regression line. The corresponding allometric relation is presented in its exponential form.</figcaption></figure>

14.3.2. A brief history of allometry

In a pioneer work from 1897, Eugène Dubois described the relation that guides the evolution of brain's mass and that of the individual in a variety of mammal species [590]. He observed that brain is smaller, relatively to the their mass, in bigger animals. He then derived an adequate expression for this relation, such as $e = c s^r$, where e is the brain's mass, s the body mass and c and r two coefficient that define the relation, with r close to 1/2, justifying the relative decrease in brain's mass that he observed. As far as we know, this represents the first mathematical expression of an allometric law, years before this term was even coined as it.

It is in 1907 that Lapicque [591] had the idea to transform Dubois’ relation in a log-log dependency, giving a straight line representation in logarithmic coordinates that is now familiar to us, cf. Figure 14.8. At that time, this work was purely descriptive and empirical. However, biological and ecological data started to accumulate in the following years that led, mainly in animal species, to a variety of scaling laws. Thus, the ubiquity of allometric relations in every ecological discipline [592] raised the question of the nature of the biological mechanisms underlying their observation.

In parallel, the question of the emergence of forms in living organisms arose in the literature. One of the major works at that time came from the Scottish naturalist D'Arcy Wentworth Thompson, whose main contribution came from his book *On Growth and Form*, first published in 1917 [593]. In this publication, he adopted the – still debated – thesis that the living systems as we know are submitted, in addition to the process of natural selection, to the physical laws of nature that can modify, transform and adapt their form and their path of development i.e., their growth [594]. This reference publication paved the way to the new disciplinary research field of biomathematics and, even in present times, is still considered as a major contribution to this field [595].

However, the D'Arcy Thompson's approach has not been accepted by the whole community, and the debate is still vivid more than a century after the publication of the first edition of his work [596]. Indeed, D'Arcy Thompson was not entirely convinced by the pure Darwinian approach that dominated the field of developmental biology in his time. Although a strong Darwin's admirer [597], he rather considered that the paths of development of the organisms were not dictated purely by acquired mutations and hard-encoded routines. At the contrary, he was convinced that these paths could only follow a number of sequences, a series of schemes that, following the laws of physics and chemistry, would allow for the formation of the variety of shapes and developments observed in nature [595, 596]. Critics emerged about his teleological – in some ways [597] – conception of evolution, or at least of emergence of form. In essence, his work was one of his time, and his theories of forces of development were not supported by the genetic and molecular
Σ Math box 13.C The value and cost of information for growing populations

To see how uncertainty may be quantified by Shannon, first consider a model where $f(R, E)$ is non-zero only for one phenotype $R$ best adapted to each particular environment $E(R)$. As seen in Box 2, the optimal strategy in the long-term is proportional betting, $u(R(E)) = p(E)$. To quantify the cost of uncertainty, it must be compared to a situation where full information is available, in which case all the cells can systematically adopt the optimal phenotype, leading to an ideal growth rate $\Lambda^{**} = \sum_E p(E) \ln f(R(E))$. The cost of uncertainty is $\Lambda^{**} - \Lambda^* = -\sum_E p(E) \ln p(E)$, which is nothing but the Shannon entropy of the environment $H(E)$. This has a simple interpretation: the more unpredictable the environment, the larger its entropy and the lower the maximal growth rate of the population.

The reasoning can be extended to the presence of partial, modeled by $p(S|E)$. The case of perfect information is indeed the limit case where $S = E$. The optimal strategy with partial information is a generalization of proportional betting that takes into account $S$ and the difference of growth rate is now given by the mutual information $I(S, E)$ (Problem 13.1). The mutual information is minimal when the signal $S$ is uncorrelated to $E$, in which case $I(S, E) = 0$, and maximal in presence of perfect information, in which case $I(S, E) = H(E)$ [415].

These results were first derived by Kelly [413]. They have been generalized to more general forms of $f(R, E)$ as well as to more general environmental processes in the context of financial investment in which case the cost of uncertainty and value of information are no longer equal but bounded by information theoretic quantities [415, 416]. This is illustrated in Problem 13.2 with an extension of the model of persistence presented in the main text. Information is generally costly as it implies producing and operating an accurate sensor, which may come at the expense of growth rate. Taking into account this cost introduces a trade-off between the cost and value of information that may justify an imperfect sensor, or even explain an absence of sensor (Problem 13.1). This trade-off has for instance been invoked to explains that bacteria subject to infrequent periods of antibiotics evolved to stochastically switch their phenotype rather than to sense the presence of antibiotics [417].

While the problems of information processing in biology and in finance share many analogies, it is also important to recognize an important difference: in biology, information processing is distributed at the level of each cell, which may perceive different signals, while in finance, information is processed by an investor who centralizes the information. The value of information is bounded by information theoretic quantities only in the second case, or more generally when the same common information is available to all the cells [416]. If information processing is stochastic at the single cell level, the value of information is effectively higher (Problem 13.3).

A general inequality characterizes this trade-off mathematically for an arbitrary number of horses. For Kelly's gambling model with fair odds defined in the box 13.B, it has the form

$$ \sigma_W \geq \frac{W}{\sigma_q}, \qquad (13.10) $$

where $\sigma_W$ is the volatility of Kelly's model, $W$ the average growth rate (the equivalent of $\Lambda$) and $\sigma_q$ is the standard deviation of a distribution, $q(x)$ defined by $q(x) = r(x)/p(x)$. This distribution compares the probability of races outcomes described by $p(x)$ with the risk-free strategy described by $b(x) = r(x)$, for which $\sigma_W = W = 0$ [421]. Recently, a similar bound has been derived for other well-known financial models such as the Black-Scholes and the Heston models [422].

Let us now illustrate the implications of this trade-off for a biological population using a simple bet-hedging model with only two phenotypes. Individuals in the population can switch from phenotype $A$ to phenotype $B$ with a transition probability $\pi_1$, and with probability $\pi_2$ from $B$ to $A$, assuming no sensing. The population grows in an environment that fluctuates between two values 1 and 2. We denote the population vector, which describes the number of individuals in each phenotype at a given time $t$ by $N(t) = (N_A(t), N_B(t))^T$, where $T$ denotes the transpose. The subpopulation of individuals with phenotype $A$ grows when placed in the environment $i$ with the growth rate $k_{Ai}$, while the other subpopulation with phenotype $B$ grows with rate $k_{Bi}$. The population is assumed to be large, there is no population noise, the dynamics of the system is deterministic in each separate environment. The population dynamics of the model can be described by the vector equation:

$$ \frac{d}{dt} \mathbf{N}(t) = M_{S(t)} \mathbf{N}(t), \qquad (13.11) $$
knowledge that has since been accumulated [596].

D'Arcy Thompson was an author of his time. He paved the way, with others developmental naturalists, to numerous concepts in biomathematics that influenced a number of past [598] and present works, as discussed in Section 14.1.

But D'Arcy Thompson was also an author among his peers. Motivated by his conception of developmental shaping forces, he started to correspond with a younger British naturalist named Julian Huxley, who will later forge a prolific international career as a biologist and science advocate, although carrying with him some controversies that are beyond the scope of this chapter [599].

The scientific correspondence started slightly after one of Huxley's major publications, dated from 1924. In this article, Huxley studied the dynamics of growth of chelae in a crab species whose individuals possess one small and one large chela [600]. What seems at first a highly specific topic is enlarged by the idea to measure the mass of the chelae *relatively* to the mass of the individual. Following the steps of Dubois and Lapicque, Huxley weighted around 400 specimens of crab and plotted in a logarithmic scale the mass of the large chela against the total weight of the animal minus the weight of the large chela. He then observed that the experimental data could be joined by a straight line in this logarithmic plot.

The originality of Huxley's work resides in his interpretation of the results that he obtained. He noticed that the slope *k* of the regression line remained larger than one, in accordance with the observation of the relative larger i.e., heterogonic growth [601] of the chela compared to the growth of each individual. He then provided a proposed mechanism for this relative growth: the rate of cellular division in the chela is larger than the one in the rest of the body, more precisely in a *k*:1 ratio [600].

With this – still emergent – mechanistic approach, Huxley provided for the first time a simple method for deciphering heterogonic growth of a characteristic, that will be observed as a straight line of slope *k* > 1 when plotted against the normalized mass of the individual in logarithmic coordinates.

Finally, the works of Lapicque, Dubois, D'Arcy Thompson and all their contemporaries emerged in 1936 in a joint paper between Huxley and a younger scientist, Georges Teissier, in which they agreed for the terminology of *allometry* and the associated law that is now famous $y = bx^\alpha$ [602].

Altogether, this brief section on the historical emergence of the allometric concept in ecological sciences depicts a vibrant and active research theme, developed in the late 19th century, which extends the Darwinian concept of natural selection towards the emergence of growth, form and function.

However, the reader will notice that the allometric approach of these times is still largely descriptive, with limited causal explanations of the nature of the scaling coefficients and the putative mechanisms that drive their behavior.

### 14.3.3. Allometry: a mechanistic approach

Many years later, a possible approach that compensates for this lack of mechanistic causality would be found in the work of West, Brown and Enquist (WBE), published in 1997 [587]. In this major article, the authors focused on the allometries in metabolic properties that have been described in the past decades, with the aim of developing a new mechanistic framework that would explain these allometries i.e., be able to derive the allometric exponents for the numerous physiological properties at stake here.

The question of the existence of a general allometry for the metabolic rate of the living beings is a thrilling question. This would imply that all the organisms, from the tiny bacteria to the massive trees or mammals, do possess shared mechanisms of energy expenditure that would reflect on the presence of a common exponent all over the different orders of magnitudes among the species. Furthermore, the exponent should reflect somehow, by its value, the nature of the energetic mechanisms, and thus could be derived by a comprehensive modelling approach.

WBE answer positively to these strong hypotheses, and developed a structured approach that focuses on the modelling of energy and mass fluxes in biological networks – cardiovascular and respiratory systems for example – which they
<figure><img src="image_122.png" /><figcaption>Figure 13.3: Pareto diagram showing the growth rate W versus the fluctuations measured by the standard deviation of the growth rate σ<sub>W</sub> (which is the volatility for this model) in the simple case that only two horses are present. The curve can be obtained by varying a risk aversion parameter α, which enters in the definition of an objective function (see Problem 13.4 for details). The point of maximum growth rate (red square) corresponds to Kelly's strategy and divides a trade-off branch (blue solid line) from a non-trade-off branch (red solid line) (adapted from [421]).</figcaption></figure>

with matrices

$$
M_{S_1} = \begin{pmatrix} k_{A1} - \pi_1 & \pi_2 \\ \pi_1 & k_{B1} - \pi_2 \end{pmatrix} \quad \text{and} \quad M_{S_2} = \begin{pmatrix} -\pi_1 + k_{A2} & \pi_2 \\ \pi_1 & k_{B2} - \pi_2 \end{pmatrix}. \qquad (13.12)
$$

The finite time averaged population growth rate is defined as

$$
\Lambda_t = \frac{1}{t} \ln \frac{N(t)}{N(0)}, \qquad (13.13)
$$

in terms of the total population $N(t) = N_A(t) + N_B(t)$, and the long term population growth rate is

$$
\Lambda = \lim_{t \to \infty} \Lambda_t. \tag{13.14}
$$

This optimal long term growth rate Λ can be obtained analytically in this model [423], but approximations
are needed to evaluate the fluctuations of the growth rate, which is the equivalent of the volatility σ² of
Eq. 13.9 [424]. One can then study the trade-off that exists between the average growth of the population
(either measured instantaneously or over a long time) and the fluctuations of the growth rate, using the
same Pareto plot used for Kelly's model in Figure 13.3. This "efficient frontier" is shown in Fig. 13.4, and as
in the case of Kelly's model, in the region of fast growth rate, it is advantageous for a population to trade
growth for less risky fluctuations. In this model, σ² correlates with the probability that the population
N(t) goes below a certain threshold, where the population is considered as extinct. The probability of
extinction is not monotonic along the Pareto front, which explains why in the region of low growth rate, it
is more advantageous to prioritize instead the increase the growth rate to avoid extinction.

In the context of ecology, besides the probability of extinction, a quantity of interest is the chance for a
population to grow from rarity in the presence of other species. In agreement with the above trade-off,
it was found that this chance can not be predicted only from the mean growth rate, and that the mean
growth rate and its variance should be both used for such a prediction [425]. In summary, the similarity of
the Pareto plots (called efficient frontier in finance) obtained in Kelly's model and in models of biological
populations in fluctuating environments [424], and evidences from various works in ecology, suggest that
consider as the common ground for all the species [587].

The hypotheses of WBE are of strong nature, and have been discussed largely in the literature (see for example [603, 604]). Although this important – and still open – debate lies beyond the scope of this chapter, it appears important to emphasize that the WBE approach created a mechanistic, mathematical framework for the study of allometric relations that, somehow, acted as a bridge between the traditional descriptive allometry and the modern mechanistic approach.

### 14.3.4. Allometric relations for the respiratory system

As far as the respiratory system is concerned, the model of WBE appears to act as a promising framework for the study of the allometric relations of this system [605].

Indeed, the lungs of mammals are built as a network of mass and energy transfer, as described before, and share morphological and functional properties, raising the question on whether the previous results for human can be extended or not to all mammals. These properties are known to be dependent on the mass *M* of the mammal with allometric scaling laws [586, 587, 602, 606, 607]. Furthermore, the physics of ventilation, and hence its control, is linked to the geometry of the lung. Consequently, the morphological differences among mammals also affect the control of ventilation.

First, our gas transport model for the human lung presented in the previous section can be slightly modified to be valid for all mammals. Indeed, we know that the lungs of mammals share invariant characteristics [574] such as the tree-like structure with bifurcating branches and the decomposition into two parts: the bronchial tree and the acini. The derivation of a lung model that depends only on mammal mass requires to relate explicitly the morphological parameters involved in our model such as the tracheal radius and length, with the animal mass. We used the datasets from [587].

The oxygen transport and exchange now occur in the idealized lung that has been generalized to fit any mammal. The transport of oxygen in the mammals lung is still driven by the tree phenomena: convection by the airflow, diffusion and exchange with blood through the alveoli walls. Hence, in each airway, the partial pressure of oxygen follows the convection-diffusion-reaction Equation (14.2) previously defined. The exchange coefficient *β* is dependent on the mammals mass since it depends on the radius of the alveolar duct which follows an allometric law.

Finally, we search for the minimum of the total energetic cost of breathing *P* relatively to the tidal volume *V*<sub>T</sub> and the breathing frequency *f*<sub>b</sub> with a constraint on the oxygen flow to blood that has to match the oxygen flow demand *V̇*<sub>O₂</sub>. Since allometric scaling laws for oxygen flow demands for mammals at basal, field and metabolic rates are available in the literature [586, 608, 609, 610], we can compute the desired oxygen flow *V̇*<sub>O₂</sub> depending on the mammal mass and on the metabolic regime.

Our model predicts that breathing frequencies and tidal volumes follow indeed allometric scaling laws. Furthermore, these laws can be derived in three different metabolic regimes: basal metabolic rate (BMR), field metabolic rate (FMR) and maximal metabolic rate (MMR), as seen in Figure 14.9,

$$ 
\begin{align*}
f_b^{\text{BMR}} &\approx 0.61 M^{-0.27} \text{ Hz, } V_T^{\text{BMR}} \approx 6.1 M^{1.04} \text{mL,} \\
f_b^{\text{FMR}} &\approx 1.17 M^{-0.31} \text{ Hz, } V_T^{\text{FMR}} \approx 11.8 M^{0.97} \text{mL,} \\
f_b^{\text{MMR}} &\approx 1.37 M^{-0.17} \text{ Hz, } V_T^{\text{MMR}} \approx 29.7 M^{1.01} \text{mL.}
\end{align*}
$$

It predicts exponents that are in accordance with the values observed in the literature. Indeed, breathing rate at BMR has been estimated to follow the law $f_b^{\text{BMR}} \approx 0.58 M^{-1/4}$ Hz [611] and tidal volume to follow the law $V_T^{\text{BMR}} \approx 7.14 M^1$ mL [587, 612]. At other metabolic rates, less data is available in the literature except for the breathing rate of mammals at MMR, estimated to follow the law $f_b^{\text{MMR}} \approx 5.08 M^{-0.14}$ Hz [613]. The validation of our model at both minimal and maximal metabolic regimes suggests that its predictions should be coherent whatever the regime, in the limit of the availability of its input parameters. This indicates that the mechanical power spent for ventilation might
<figure><img src="image_123.png" /></figure>

Figure 13.4: Pareto diagram showing the population growth rate versus the fluctuations of that growth rate in a simple model of a biological population evolving in a stochastic environment with no sensing according to Eq. 13.11 [424]. (A) The time scale of environment fluctuations is comparable to that of phenotypic fluctuations. (B) The probability $P_{\text{ext}}$ that the population goes below a certain extinction level versus the risk aversion parameter $\alpha$ which measures the distance along the Pareto plot. Colored bullets represent different points on the Pareto front (adapted from [424]).

the trade-off discussed here is broadly applicable in various fields ranging from biology and ecology to economics.

## 13.5 Strategies in correlated environments

So far we considered two time scales: the time scale at which phenotypic changes occur and at which instantaneous growth is defined ($t = 1$ in our discrete-time model, which may be taken to correspond to one generation), and the longer time scale $t \sim (\sigma/\Lambda)^2$ beyond which population growth is effectively exponential, with growth rate $\Lambda$. We saw that the choice of an optimization criterion depends fundamentally on the time horizon relative to these time scales.

Additional time scales are relevant when environmental states are correlated in time, for instance through a Markov chain $P(E_t|E_{t-1})$. This is for instance the case if conditions of high nutrient or high stress extend over several generations. As a consequence, strategies $u(R_t|S_t, R_{t-1})$ that depend on past internal states $R_{t-1}$ in addition or instead of externally driven signals $S_t$ may become valuable, since the fact that phenotype $R_{t-1}$ survived in environment $E_{t-1}$ indirectly carries information on the current environment $E_t$. We may then recognize that $R_t$ plays two distinct roles: on one hand, it determines survival and growth via $f(R_t, E_t)$ and, on the other, it provides information to determine the next state $R_{t+1}$ via $u(R_{t+1}|S_{t+1}, R_t)$. This corresponds to the fundamental distinction between phenotype and genotype in biology: the genotype $\gamma$ is what is transmitted from one generation to the next while the phenotype $\phi$ is what determines instantaneous growth. Formally, $R_t = (\phi_t, \gamma_t)$ with $f(R_t, E_t) = f(\phi_t, E_t)$ and $u(R_t|S_t, R_{t-1}) = u(R_t|S_t, \gamma_{t-1})$, by definition of $\phi_t$ and $\gamma_t$. The “central dogma” of molecular biology states that information flows from the genotype to the phenotype but not reciprocally, which corresponds here to assuming that $u(\phi_t, \gamma_t|\gamma_{t-1})$ factorizes as $d(\phi_t|S_t, \gamma_{t-1})h(\gamma_t|\gamma_{t-1})$, where $d(\phi_t|S_t, \gamma_{t-1})$ may be interpreted as a developmental kernel and $h(\gamma_t|\gamma_{t-1})$ as an inheritance kernel, with no dependence on $S_t$ (no Lamarckism). The mathematical framework that we introduced can be used to study to which extent this particular decomposition is indeed a good “strategy” [426]. The answer generally depends on the nature and amplitude of the environmental fluctuations.

Similarly, the model can be analyzed to understand the conditions under which it is advantageous to introduce phenotypic variations that are not transmitted – as in bet-hedging – versus genotypic variations that are transmitted – as with genetic mutations. Stochasticity may indeed be introduced either in the mapping from $\gamma_{t-1}$ to $\phi_t$ or the mapping from $\gamma_{t-1}$ to $\gamma_t$, or in both of them – a problem with no equivalent in finance. This is illustrated in Fig. 13.5 with a simple solvable model showing how the optimal strategy
<figure><img src="image_96.png" /><figcaption>Figure 14.9: Predicted ventilation frequency (s⁻¹ – left) and tidal volume (L – right) as a function of the mammal mass (kg – log-log scale) at different metabolic regimes. BMR: Basal Metabolic Rate, FMR: Field Metabolic Rate, MMR: Maximal Metabolic Rate.</figcaption></figure>

have driven the selection by evolution of the ventilation patterns.

The idealized representation of the bronchial tree and of the exchange surface used in this study accounts for five core characteristics common to all the mammals lungs, as identified in the literature [574, 575, 581, 587, 614]: a bifurcating tree structure; an homogeneous decrease of the size of the bronchi at the bifurcations; the size of the trachea; the size of the alveoli; and the surface area of the exchange surface. These characteristics are the main determinants for the tuning of the ventilation in order to minimize its energetic cost. This indicates that once the metabolic regime is fixed, the morphology of the lung is probably the primary driver of the physiological control of ventilation.

We tested this hypothesis by altering, in our analysis, the allometric scaling laws related to the geometry of the lung. We observed corresponding alteration of the laws predicted for tidal volumes and breathing frequencies. Since morphology itself has probably been selected by evolution in order to minimize the hydrodynamic resistance in a constrained volume [575], morphology and ventilation patterns are intertwined together in order for the lung to function with a low global energetic cost i.e., a low hydrodynamic resistance *R* and a low ventilation cost *P*(*V*<sub>T</sub>, *f*<sub>b</sub>) that also depends on *R*.

Interestingly, our representation of the lung does not account for interspecific differences known to exist between the lungs of mammals, such as different degrees of branching asymmetry, monopodial or bipodial lungs, etc. [615, 616, 617, 618].

As in the human lung, the transport of gases in the mammalian lung relies on the two major processes of diffusion and convection. We know that, in humans, the diffusive transport in the alveolar ducts is submitted to a physical phenomenon called the *screening effect* [577, 619]. Indeed, as gas exchanges occur through the alveoli walls lining the alveolar ducts, the diffusion can transport the respiratory gases only on a limited range of generations. This range depends on the physico-chemical properties affecting the diffusion of the gas in the alveolar air and through the alveolo-capillary membrane. This range has been estimated to be of about four generations for oxygen and one for carbon dioxide [577] in humans. The description of the screening effect in mammals requires several additional hypotheses.

Because of the screening effect, the alveolar ducts far from the convection-diffusion transition get only a small diffusive oxygen flow, as most of the available oxygen has been captured by the alveolar ducts closer to the transition. In these deep parts of the acini, the oxygen partial pressure gradient between the deoxygenated blood and the alveolar ducts, which drives the oxygen capture by blood, is low. Carbon dioxide is mostly evacuated from the alveolar ducts very close to the transition: they are refilled by carbon dioxide too quickly for the deeper ducts to be drained of gas by diffusion. Hence, the ducts far from the transition cannot be relieved of the carbon dioxide and the exchange with blood in these ducts is low. As a consequence, the deeper part of the exchange surface is not available for the exchanges. The location of the transition between convective and diffusive transport of the respiratory gas drives the magnitude of the screening,
<figure><img src="image_124.png" /></figure>

Figure 13.5: Example of optimal strategies in correlated environments – (A) We consider here a model where the environment $E_t$ is a continuous variable following a Gaussian process, $P(E_t|E_{t-1}) = \exp(-(E_t - aE_{t-1})^2/(2\sigma_X^2))/(2\pi\sigma_X^2)^{1/2}$ with two parameters $a$ and $\sigma_X^2$ that control the overall amplitude of the fluctuations $\sigma_E^2 = \sigma_X^2/(1-a^2)$ and their time scales $\tau_E = -1/\ln a$, as illustrated by the different time series. (B) An individual inherits a genotype $\gamma_{t-1}$ which determines its phenotype $\phi_t$ with probability $d(\phi_t|\gamma_{t-1}) = \exp(-(\phi_t - \gamma_{t-1})^2/(2\sigma_D^2))/(2\pi\sigma_D^2)^{1/2}$ where $\sigma_D^2$ thus represents phenotypic noise. $\gamma_{t-1}$ also determines the genotype $\gamma_t$ of the progeny with probability $h(\gamma_t|\gamma_{t-1}) = \exp(-(\gamma_t - \gamma_{t-1})^2/(2\sigma_M^2))/(2\pi\sigma_M^2)^{1/2}$ where $\sigma_M^2$ thus represents mutational noise. The number $\xi$ of offsprings is a random variable whose mean $f(\phi_t, E_t) = k \exp(-(\phi_t - E_t)^2/2)$ depends on the phenotype $\phi_t$ as well as the current state $E_t$ of the environment. A population of such individuals grows with a long-term growth rate $\Lambda$ that can be computed analytically [426]. (C) Values of $\sigma_D^2$ and $\sigma_M^2$ that optimize $\Lambda$ define four phases as a function of the environmental parameters $\tau_E$ and $\sigma_E^2$. For nearly constant environments, the optimal strategy is to maintain constant phenotypes ($\sigma_D^2 = 0$) and genotypes ($\sigma_M^2 = 0$) ("no variation"). For strongly varying but poorly correlated environments, the optimal strategy is to introduce phenotypic variations ($\sigma_D^2 > 0$) but no genotypic mutations ($\sigma_M^2 > 0$) ("phenotypic switching"). For highly correlated environments, the optimal strategy is instead to introduce genotypic mutations ($\sigma_M^2 > 0$) while canalizing the phenopype ($\sigma_D^2 = 0$) ("inherited variations"). A phase also exists where both types of variations are beneficial ("mixed"). This model thus identifies environmental variations for which bet-hedging (phenotypic switching) is expected to evolve, namely variations of environmental of sufficient amplitude but with limited temporal correlations across generations.

depends on the nature of the fluctuations of the environment. In particular, bet-hedging strategies where stochasticity is purely phenotypic are found to be optimal for environmental fluctuations of sufficient large amplitude but low temporal correlations from one generation to the next.

Historically, the notions of genotype and phenotype were introduced much before the molecular mechanisms that underlie them were uncovered. In general, the genotype, defined as inherited information, should not be confused with the notion of genetic information: along with DNA, a range of epigenetic states, including metabolic states, are also transmitted from cell to cell which represent genotypic information. In other words, the physiological state of a cell, which we analyzed in most of this book from the standpoint of a phenotype determining current growth, may also represent valuable genotypic information for future generations.

## 13.6 Concluding remarks

We presented optimal strategies that biological populations may exploit for coping with uncertain environments and drew analogies with problems of gambling and financial investments. Optimality assumes a measure of performance which, however, is not readily defined when environments are changing stochastically. In particular, the time scale over which the problem is considered is critical. This difficulty has led to multiple debates over the concept of fitness in biology which partly mirror those over the concept of
<figure><img src="image_97.png" /><figcaption>Figure 14.10: Localization in terms of lung generation index of the conductive zone and of the exchange surface (acini) as a function of the mammal species mass (kg). Both the green line (rest regime – left) and the red line (maximal exercise regime – right) represent the transition from a transport of the respiratory gas by convection to a transport by diffusion. Adapted from [605].</figcaption></figure>

and this transition depends on the geometry of the airway tree and of the ventilation regime.

The screening phenomenon in mammals has been studied mathematically in [605]. Within the framework of the models hypotheses, the authors show that the number of conductive airways $N_{conD}$ and the number of alveolar ducts $N_{ad}$ follow allometric scaling laws:

$$N_{conD} \propto N_{ad} \propto M^{\frac{7}{8}}$$

Additionally, they show that the number of airways $N_{conV}$ in which the gases are transported by convection also follows an allometric scaling law. This law depends on the ventilation regime:

$$N_{conV} \propto \begin{cases} M^{0.56} & \text{if } M < 150 \text{ kg} \\ M^{0.405} & \text{if } M \ge 150 \text{ kg} \\ M^{0.63} & \text{at maximal exercise} \end{cases} \quad \begin{array}{l} \text{at rest} \\ \text{at maximal exercise} \end{array}$$

These equations translate into linear relationships in terms of $log(M)$, as shown in Figure 14.10. Rest regime is represented on the left plot and maximal exercise regime on the right plot.

The figure indicates that, at rest regime, the small mammals use their lung very efficiently, as only a few of their acini generations are fed by diffusion, as indicated by the green curve in Figure 14.10. Hence, the screening effect in small mammals is weak. However, this suggests that they have few reserve for increasing their metabolism at exercise [577, 605]. As suggested by the red curve on the right plot in Figure 14.10, the shift of the transition between convection and diffusion to deeper generations does not increase significantly the available exchange surface.

To the contrary, large mammals are submitted to large screening effects at rest regime, and a large part of their exchange surface is not used. However, during exercise, the shift of the transition towards a deeper lung generation allows to recruit a significantly larger exchange surface.

### Box 14.D: Convection-diffusion transition and allometry

It is to be noted that the predictions of our model for the localization of the convection-diffusion transition in idealized lungs lead to good estimations of the allometric scaling laws for tidal volumes and breathing frequencies,
utility in economics.

While the analogy with finance is instructive, its limitations should also be kept in mind. Most importantly, the states that individuals of a biological population adopt are not centrally controlled by a gambler or an investor. This raises a question that is absent in finance but central in evolutionary biology: is a strategy that is optimal for the population but detrimental to some of its members – as for instance the persister cells that “sacrifice” their current growth for the sake of future growth – evolutionary stable? A strategy that is optimal for a population may indeed never be achieved through evolution as natural selection at the individual level may favor non-cooperating individuals – an issue known as a “conflict between levels of selection” which implies that a strategy may be optimal at the population level but not evolutionarily stable. To address this question, we may extend our model to treat strategies as variables that are themselves subject to evolution (Problem 13.5). For the model discussed in this chapter, the results show that strategies that optimize the long-term growth rate are indeed evolutionarily stable (but this is no longer necessarily the case when considering, for instance, sexually reproducing populations [427]).

The same extension of the model to evolving strategies shows that knowledge of the statistics of the environment ($p_d$ for our example) is not required a priori but can effectively be learned through an evolutionary process. This solves a problem that appears also in gambling and finance where the statistics of the environment must be inferred from past experience. The question has been particularly studied in finance, where optimal learning strategies known as universal portfolios have been proposed [428]. In the simpler case of Kelly's model, the gambler may for instance record previous race results and use them together with Bayesian inference to predict the probability of the race outcomes [429]. With biological populations, however, learning must be performed at the individual level. One theoretical proposal that goes beyond random mutations is for instance that biological populations may use a reinforcement mechanism akin to Hebb's rule in neural learning [430].

Finally, we note that the models that we presented rely on a strongly simplifying assumption: the environmental changes occur independently of the population. In fact, the environment is often also changing as the population grows, for instance through the consumption of nutrients. Even more generally, the environment may comprise other individuals from the same or other populations with which they may interact. This ecological dimension is the subject of other chapters.

## Recommended readings

**Persistence** This reference describes the phenomenon of bacterial persistence. Nathalie Q Balaban, Jack Merrin, Remy Chait, Lukasz Kowalik, and Stanislas Leibler. Bacterial persistence as a phenotypic switch. Science, 305(5690):1622–1625, 2004. doi: 10.1126/science.1099390

**Information theory for decision making under uncertainty** This book is a classic text on the use of information theory in problems from finance. Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. John Wiley & Sons, Hoboken, 2005. doi: 10.1002/047174882X

**Model for information sensing** The Kussel-Leibler model is one of the first to incorporate information theory into cellular behavior and signaling. Edo Kussell and Stanislas Leibler. Ecology: Phenotypic diversity, population growth, and information in fluctuating environments. Science, 309(5743):2075–2078, 2005. doi: 10.1126/science.1114383

## Problems

### Problem 13.1 Kelly strategy with partial information

In analogy with Kelly's problem of betting on horse races, assume that different environments $E$ occur with independent probabilities $p(E)$ at each generation with a single phenotype $R = E$ permitting growth by a factor $f(E)$. In absence of any information, the optimal strategy $u(E)$ for long-term growth is proportional betting, $u(E) = p(E)$ (Box 13.B). Now assume that an information $S$ is available to each member of the population that relates to $E$ through a transition probability $q(S|E)$, i.e., $q(S|E)$ is the probability of perceiving $S$ given $E$.
indicating that the morphological parameters included in our model might drive primarily the control of ventilation.

Box 14.E: Interlude – Cost and energy

The allometric relationship found applies to the pulmonary organ. This is a crucial link in muscular activity, and therefore in locomotion or any activity requiring an effort, even moderate. As such, its properties must also be present during physical exercise. A useful quantity, based on oxygen consumption $\dot{V}_{O_2}$ and frequently used in the literature, is the Cost of Oxygen Transport (COT). This corresponds to the ratio $\dot{V}_{O_2}/v$ with $v$ the locomotion velocity. Using the correct metabolic conversion factor COT is the energy dissipated per unit length. It is known empirically that COT shows a local minimum corresponding to an optimal situation in which the minimum energy is dissipated per unit length. Building on this property, Tucker in 1975 [620] noted that this minimum follows distinct allometric laws according to the major locomotion families, runners, swimmers and fliers, see Figure 14.11, left.

<figure><img src="image_98.png" /><figcaption>Figure 14.11: Left is the COT defined here as the ratio $P/(Mv)$, with $P$ the power production and $v$ the velocity as a function of the body mass $M$ for several species (adapted from [620]). Green are swimmers, red are fliers, black are runners, blue are engines designed by engineers. Continuous lines correspond to linear fits on data shown with filled markers. Right, top is oxygen consumption $\dot{V}_{O_2}$ of a horse plotted against the speed $v/v^*$ for walk (red stars), trot (blue dots), and gallop (green squares), and their fits with our modelling. Bottom is COT for the same set of data. The three gaits data are normalized by the muscle fiber ratio leading to a unique master curve.</figcaption></figure>

Based on the model proposed in the Interlude – Energy, it has been demonstrated [621] that a living system can be described as a collection of $N$ identical, standard, muscle units operating in parallel. Then the COT expression becomes:

$$ \text{COT} = \frac{N}{N_H} \left( a_0 k + r_M k^2 v + \frac{b}{v} \right) \qquad (14.3) $$

with $a_0 k$ a constant, $r_M$ a dissipative term and $b$ the basal consumption i.e., out of effort. This last three parameters describing the standard muscle unit. We are then allowed to derive the minimum of the COT as an intrinsic property of energy conversion machines, $COT_{min} \propto \sqrt{r_M b}$. It is found independent of the number of standard muscle fibers involved in the effort. Thus, effort is a combination of the number $N$ of standard muscle fibers used and their characteristics $b$ and $r_M$. The parameterization of the standard muscle fiber depends on the specific implementations for an organism. It can be expected to be identical for a single animal. We have carried out this work in the case of the horse, which exhibits three well-differentiated gaits: walk, trot and gallop (see Figure 14.11, right). We show that the COT curves, or equivalently $\dot{V}_{O_2}$, of the different gaits can be found using $N$ as the only adjustable parameter, leaving the muscle fiber parameters unchanged.
(a) Show that the long-term growth rate can be written in the form

$$ \Lambda = \sum_{S} p(S) \left( \sum_{E} p(E|S) \ln(f(E)u(E|S)) \right) \qquad (13.15) $$

where $p(S)$ is the probability to perceive $S$ averaged across all environments and $p(E|S)$ is the probability that environment is $E$ given that $S$ is perceived. Write $p(E|S)$ as a function of $p(E)$ and $q(S|E)$.

(b) Justify that the optimal strategy is $u(E|S) = p(E|S)$.

(c) Compare the optimal long-term growth rate in presence of information to the optimal growth rate in absence of information and show that the difference is given by the mutual information

$$ I(E, S) = \sum_{E,S} q(S|E)p(E) \ln \frac{q(S|E)}{p(S)} \qquad (13.16) $$

The mutual information $I(E, S)$ therefore quantifies the value of information $S$ in this particular context.

(d) Acquiring information is generally costly. If the presence of the information channel $q(S|E)$ reduces the long-term growth rate by $c$, what are the conditions on $p(E)$ for the presence of this channel to be beneficial?

(e) The cost $c$ may be expected to depend on the precision of the sensor. Consider for instance a channel that reveals the correct environment with probability $1 - \epsilon$ and otherwise does not reveal anything (so-called erasure channel). Given a cost $c(\epsilon)$ that increases when $\epsilon$ decreases, which value of $\epsilon$ provides an optimal trade-off between the value and the cost of information?

### Problem 13.2 Value of information beyond Kelly's model

Consider the model of bacterial persistence introduced in the main text where cells can adopt two phenotypes, one growing irrespectively of the environment and the other growing only in absence of antibiotics.

(a) Express the long-term growth rate $\Lambda$ in presence of an information $S$ modeled by an information channel $q(S|E)$.

(b) What is the optimal strategy given $S$?

(c) Show by comparing to a situation with no information that the value of information can be strictly lower than $I(S, E)$.

### Problem 13.3 Stochastic sensing at the level of individual cells

In the two previous problems, the information $S$ is assumed to be common to each member of the population. Here we assume instead that each individual has its own sensor $q(S|E)$ so that $S$ may differ from one individual to the next.

(a) Justify that in this case the long-term growth rate takes the form

$$ \Lambda = \sum_{E} p(E) \ln \left( \sum_{R,S} f(R, E) u(R|S) q(S|E) \right) \qquad (13.17) $$

(b) Use the concavity of the logarithm (Jensen's inequality) to justify that the same information channel $q(S|E)$ has more value at the individual level than at the population level.

### Problem 13.4 Pareto front for Kelly's model

Let us consider Kelly's model with fair odds for two horses. Let the probability that the first horse wins be $p$, the bet and the odd on the first horse be $b$ and $1/r$.

(a) Write the expression of the mean growth rate $\langle W \rangle$, and of the volatility $\sigma_W$ for this problem. Show that there is a risk free strategy when $b = r$.
As muscle is the most common means of producing power in animals, the typical behavior described here should be found in the most general way, without barriers between species, genera or classes. Of course, muscular implementation is specific to each animal, constrained by its own characteristics (intensity of effort, size, etc.), which suggests the origin of the observed scaling laws.

Through this short introduction to allometry of constrained organs, we started to decipher the latent mechanisms of development of a constrained organ inside a class of organisms. The example of the lung is emblematic: how a complex and central organ can develop, specialize and evolve to fulfill the needs of organisms, while sharing among species its particularities, and efficiency.

## 14.4. Conclusion and perspectives

Biological optimization, making the most effective use of limited resources within a set of given constraints, is a multi-faceted subject that has been a source of content for countless articles and a stimulus for related discussion.

To make the optimization of biological systems more readily comprehensible, this chapter has focused attention on a single organ, the human lung, and used it as a stage on which to introduce basic principles and a canvas on which to illustrate their application.

The range of constraints, for the most part energetic or morphometric in nature, that have conditioned the development of the lung over the long course of its evolutionary history and given the mammalian respiratory system its particular shape is expansive. The characteristics of these constraints and the conditions that govern their interplay can be represented as mathematical equations that form the basis for models that describe the scale of the effect constraints have on biological systems and illuminate the magnitude of their impact.

The insights into the lung's form that these models yield also provide a more thorough understanding of its function, characterizing, for example, modulations in the regulation of respiratory ventilation that occur in response to changes in the body's state – e.g., when the body is at rest or in motion; when it is healthy or when its health is compromised.

The models are also a source of results that can be abstracted and subsequently applied to both human organs and those of other species that are larger and more complex. Considered within this broader context, they can also be seen as integral elements of much larger systems and as instances of the general allometric laws to which those systems adhere. The significance of the larger orthogenetic and phylogenetic implications that this abstraction of specific models into generalized laws carries cannot be overstated and discussion of those implications is vigorous and far-reaching. Through these discussions, many aspects of allometry have been illuminated and a deeper understanding of the complex systems that determine the ways individuals, species and systems function and interact has been achieved. Yet many of the field's underlying mechanisms and governing principles remain to be discovered.

This chapter is the prelude to a journey into a space at the intersection of biology, ecology, and mathematics that the allometric universe occupies and the fuel for the exploration of the mysteries those hidden mechanisms are waiting to reveal.

## Recommended readings

Here is presented a selected list of publications that served as a basis for the preparation of this chapter and constitute a strong corpus of references for this topic.

* For a proper introduction to respiratory physiology, in healthy and pathological conditions: John B. West, *Respiratory Physiology: The Essentials* [540].

* A reading for a deeper understanding of the lung morphometry: Ewald R. Weibel, *Morphometry of the human lung* [622] and one for the respiratory gases exchange: Ewald R. Weibel, *The Pathway for Oxygen: Structure and*
One introduces the objective function

$$J = \alpha \langle W \rangle - (1 - \alpha) \sigma_W. \qquad (13.18)$$

(b) From the optimization of $J$ show that the optimal strategy has the two branches shown in Fig. 13.3. Show that the optimal bets on these two branches are of the form $b^\pm = p \pm \gamma \sigma$, where $\gamma = (1-\alpha)/\alpha$ and $\sigma = \sqrt{p(1-p)}$.

(c) Show that the slope of the Pareto border has the form

$$\frac{d\sigma_W}{d\langle W \rangle} = \frac{\sigma}{p-b}. \qquad (13.19)$$

What happens to this slope near Kelly's point and near the risk free strategy?

### Problem 13.5 Evolution of an optimal strategy

Here we consider evolving the strategy itself.

(a) Implement numerically the model of bacterial persistence introduced in the main text for a large but finite population. To this end, consider $N$ individuals (e.g., $N = 1000$), each with an attribute $R$. For each individual, draw a random number $\xi$ of descendants, with mean $f(R, E_t)$ where $E_t$ drawn from $P(E)$ is common to all individuals. Assign a $R$ to each of these descendants with probability $u(R)$. If the total number of descendants $N_t$ is non-zero, record the ratio $N_t/N$ and re-sample at random the population to bring back its size to $N$. Show that provided that $N$ is large enough and $N_t$ does not reach 0 then $(\sum_t \ln(N_t/N))/t$ provides a good approximation to the growth rate $\Lambda$ in the limit of large $t$.

(b) Extend the model to make $u_d u(R = \text{dormant})$ an attribute of each individual. Assume that $u_d$ is transmitted from one parent to one of its offspring as $u_d = \min(1, \max(0, u_d + \mu))$ where $\mu$ is normally distributed with variance $\sigma_M^2$. Show that provided that $\sigma_M^2$ is small enough, the distribution of $u_d$ evolves to be centered around the optimal $u_d$.
* Function in the Mammalian Respiratory System [574].

* A nice thesis about (in)organic mechanisms of morphogenesis: Raphaël Clément, *Morphogénèse et développement pulmonaire* [623].

* The old but gold textbook in morphogenesis of living beings: D'Arcy Wentworth Thompson, *On Growth and Form* [593].

* On allometric relations, in general: Robert H. Peters, *The Ecological Implications of Body Size* [586] and from a modelling approach: G. B. West et al., *A general model for the origin of allometric scaling laws in biology* [587].
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Chapter 14

Strategies for cell cycle control

Mattia Corigliano, Marco Cosentino Lagomarsino, Jacopo Grilli, and Gabriele Micali

Chapter overview

○ Cells require coordination of growth and division, as well as coordination of cell-cycle progression with several essential sub-tasks, such as chromosome replication and segregation.

O Single-cell dynamics data offer correlation patterns that can be used to understand these deci-
sional processes.

○ The cell-cycle progression and cell-division decisional process can be described by continuous-time and discrete-time stochastic processes.

○ There are quantitative relationships that connect growth, cell-cycle progression, and resource al-
location.

○ There are differences and common points in the decisional processes by which single cells of dif-
ferent organisms commit to divide (sizers, adders, accumulators, titration-dilutors, etc.)

14.1 Introduction: the decision to divide illustrated through single-cell
*E. coli* data.

As nicely put by the Nobel prize winner François Jacob, “the dream of every cell is to become two cells”. Achieving this dream often requires multiple steps, such as growing by a certain size, replicating DNA, and dividing. The previous chapters have addressed cell growth as a consequence of optimization of catabolic and biosynthetic fluxes through optimally regulated resource allocation; this chapter deals with the decision to divide (and to progress the cell cycle), based on growth and other important cellular processes and cues. Clearly this decision to divide or progress the cell cycle must be based on a set of inputs (growth, production processes such as DNA replication and cell-wall biosynthesis, partitioning processes, etc.) and entails several outputs, prominently cell division, but also intermediate key cell-cycle substeps, such as initiation of DNA replication or construction of a “divisome” organelle. The questions that we will consider concern the characterization of the known aspects of this decisional process and its coupling to cell size, to cell growth, and to the chromosome cycle. We will use throughout the chapter *E. coli* as an example. This section provides a description of the main problem through an introduction to the data, based on *E. coli* bacteria. Sections 2-5 start from a mathematical toolbox of models that are useful in this context and compare them with data. Finally, section 6 describes applications to other organisms than *E. coli*.

Capturing the key processes regulating cell division is a fundamental question in biology, which remains
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<figure><img src="image_125.png" /><figcaption>Figure 14.1: Salient quantitative features of cell-division control, explained through *E. coli* data – (A) *E. coli* cells are rod-like. Within a condition they grow by increasing their length, and they divide symmetrically. Following single-cell lineages, growth in length or volume is close to exponential. (B) Size-growth plots quantify the strength of division control. For a timer, multiplicative growth quantified by $G = \log(s_d/s_0)$ is uncoupled to birth size, for a sizer, it is maximally coupled. The single-cell data show an intermediate trend. (C) Since $G = \log(s_d/s_0) = \alpha\tau$, the size-growth plot can be split into contributions correlation with birth size to growth rate (top) and/or interdivision time. The data show that *E. coli* bacteria only compensate by modulating interdivision times. (D) Two equivalent quantifications of the strength of the division control size. The intermediate control strategy adopted by *E. coli* adds a size that is independent from the initial size ("adder"). This strategy is sufficient to achieve size homeostasis.</figcaption></figure>

chromosome, how the septum and the new cell wall are synthesized. However, the vast majority of these data are based on population averages, out of which it turns out to be impossible to extract any direct and/or causal link between the different processes involved in cell growth that set cell division [431]. Today, a new generation of data has the potential to answer several open questions [431, 432, 433]. These data differ from the previous generation in the ability to measure single bacterial cells over multiple division events in controlled conditions. At the same time, the expression of a specific gene or the concentration of specific proteins of interest can be monitored using fluorescent reporters. For example, fluorescent tags on the proteins involved in replication are used to score the initiation of replication in each cell cycle. Single-cell data allow for validating mathematical models and thus bring insights into the causal link between the several processes a cell need to complete before dividing.

By following lineages of cells over multiple generations under controlled environmental conditions, scientists collected different important pieces of evidence (Figure 14.1): First, within a cell cycle, the cell size $s(t)$ is well-described by a single exponential in time<fnref n="1" /> [436, 437]: $s(t) = s_0 \exp(\alpha t)$, where $s_0$ is the size at birth, $\alpha$ is the growth rate, and $t$ is the time since cell birth.

If division occurs at time $\tau_d$, a simple relationship connects the size at division $s_d$ with the other cell properties: $s_d = s_0 \exp(\alpha\tau_d)$. All the four parameters of this equation are subject to stochasticity in time and vary across single cells, even when they grow in controlled conditions. Second, in steady growth, the size distribution of newborn cells does not change over time, an observation that is referred to as cell-size homeostasis [434]. Equivalently, cells show specific correlation patterns between size at growth and size at division, which are related to their cell-division strategy [438, 434].

Let us try to understand more in detail how single-cell correlation patterns can be used to understand cell-division behaviors. The observation of near-exponential growth immediately suggests a change of variables that is useful to formulate mathematical models and to understand how single cells control cell

<footnote n="1">¹Note that most of the studies today use cell length as a proxy for size. However, different choices are possible such as volume or mass, and the differences are not fully characterized [434, 435].</footnote>
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division. Indeed, if we can assume that growth is exponential, we can use logarithmic sizes instead of linear sizes. One robust observation, is that the elongation $G = \log(s_d/s_0) = \alpha\tau$ depends on the size at birth $s_0$ (Figure 14.1B). This allows us to generate so-called "size-growth" plots (Figure 14.1B), in which the log-multiplicative growth during a cell cycle of a single cell is plotted as a function of the logarithmic size at birth [438]. Different mechanisms of size control predict different slopes for this plot. A cell division set by a "timer", for instance, would predict no relation between $G$ and size. Since $G = \log s_d - \log s_0$, if instead $\log s_d$ were independent of the initial size, a "sizer", one would predict a slope $= -1$. The E. coli data typically fall half way in between these two predictions, a negative slope of about 0.5 (Figure 14.1B).

By noticing that the overall logarithmic growth $G$ during a cell cycle is the product of the single-cell growth rate and inter-division time ($G = \alpha\tau$), we can ask the question of which one of these variables is responsible for the correlation. This analysis disentangles the contributions to cell division control due to growth rate and inter-division timing (Figure 14.1C). In other words, the dependency of $G$ on initial size can be further decomposed on the dependency of growth rate $\alpha$ and division time $\tau$. In E. coli, when growth rate and interdivision times are plotted separately as a function of the logarithmic size at birth, the negative slope is only observed in the interdivision-time plot, suggesting that cell control size by adjusting the single-cell interdivision time rather than their single-cell growth rates. Hence, E. coli data indicate that $\tau$ does depend strongly on initial size, while the growth rate has only a weak dependency [437].

One can visualize and quantify the mutual dependencies between cell sizes and growth properties in other equivalent ways (Figure 14.1D). For example, in E. coli data, the scatter plot relating size at division in the y-axis to size at birth in the x-axis for single cells has a slope of around 1 (and once again this observation holds true for different strains and under different environmental conditions). In this plot, a slope of 0 would suggest that cells on average need to reach a threshold in size upon division, a sizer. More technically, the division size $s_d$ is independent on the initial size $s_0$ in the case of a sizer. Instead, a slope of 2 in this plot would suggest that cells on average need to wait a fixed time upon division, a timer. The observed intermediate slope of 1 can also be understood using the equivalent plot in which the added size between birth and division is used on the y-axis, studying the dependency of the added size $s_d - s_0$ on $s_0$. This latter way to plot the data is particularly popular, given that, for many datasets it shows no dependency, suggesting that adding a constant added size is the mechanism of size control effectively in place. Indeed, for E. coli the experimentally observed slope is always close to 0 [439, 440, 434], an observation that goes under the name of "adder" behavior since cells appear to add on average a constant size during the cell cycle (Figure 14.1(B,D)).

It is fairly simple to rationalize why, for exponentially growing cells, a cell division strategy based on a timer does not achieve a homeostatic size. In order to do this, we can call $q(i) = \log(s_0(i))$ the logarithmic cell size at birth of cell-cycle $i$, and look at its dynamics through subsequent cell cycles. Since $s(\tau) = s_0 \exp(\alpha\tau)$, and $\langle \alpha\tau \rangle = \log 2$, and assuming that cells divide perfectly in two halves, one immediately gets that

$$q(i+1) - q(i) = \nu(i)$$

where $\nu(i)$ is a zero-average random variable independent for each cell-cycle, arising from the size-independent fluctuations of inter-division times (hence, in technical jargon, we can model $\nu$ as a discrete-time Markovian random process). Since the jumps in logarithmic size between subsequent cell cycles are random and independent, cell size at birth makes a discrete-time multiplicative random walk, hence, within a population, the distribution of cell sizes at birth tends to get wider and wider across divisions. The following two sections will explain how size homeostasis can be achieved by size-coupled cell divisions.

## 14.2 Hazard rate approach to cell division

As we have seen in the previous section, E. coli cells grow roughly exponentially. Hence, we can describe their growth by a trajectory for size $s$ (measured as cell mass or volume) of the kind $s(t) = s_0 \exp(\alpha t)$, where $t$ is time from cell birth. While experimentally the growth rate $\alpha$ fluctuates with time, we will neglect its variability and assume for the moment that it is constant. As a consequence, the cell grows as a simple exponential function of time. We will address different hypotheses regarding this point in the later
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<figure><img src="image_126.png" /><figcaption>Figure 14.2: Illustration of the inverse hazard rate approach on data – Data from many lineages of dividing cells can be used to estimate the cumulative distribution of non-divided cells, which can also be conditioned on different variables. The drawn example refers to the case where the tested variable is the added size $s - s_0$. In this case, the formalism allows to extract mathematically the hazard rate $h_d(s - s_0)$ from this distribution. Experimental E. coli data are consistent with this adder scenario, with an hazard rate that peaks at a characteristic added size, after which the division control weakens.</figcaption></figure>

sections.

A simple way to describe the decision processes leading to division (or other cell cycle progression events) is the so-called “hazard rate” model [441, 437, 440]. In this framework, as the cell cycle progresses, each cell has a certain probability to divide, and we call $h_d$ the rate of cell division. In principle, this rate can be a function of many different internal cellular parameters, all the processes that contribute setting cell division. However, since we have in mind experiments measuring cell size versus time and recording cell divisions, the most general “empirically accessible” $h_d$ can depend on $s, t, s_0, \alpha$ with the constraint that $s/s_0 = \exp(\alpha t)$. This means that there can be at most three free parameters. We can also consider simplified models, such as $h_d = h_d(s)$ or $h_d = h_d(s, t)$. Empirically, the lack of correlation between $\alpha$ and birth size suggests a smaller role for this parameter. It is important to realize that this formalism is very powerful, as it can be applied more widely to any sub-cell cycle decision (for example, entry into a specific phase, such as initiation of DNA replication, mitosis, etc.), and to measurements of different relevant cell-cycle processes (for example chromosome configurations or the expression of cell-cycle proteins or other factors), which the hazard rate may depend on.

Given a model for the hazard rate, we are interested in the cumulative probability $F(t|s_0, \alpha)$ that a cell born at $t=0$ has not divided at time $t$, given that its initial size is $s_0$ and its exponential growth rate $\alpha$. Box 14.A discusses the mathematical formalism to obtain this probability.

The considerations we made so far are sufficient to produce “forward models” where a hazard rate is assumed, and one explores its consequences on the division dynamics. The simulation of such a model is straightforward. For each discretized time increment $dt$, the cell will grow by the prescribed dynamics $s(t)$ and will divide with hazard rate $h_d$. If a division occurs, the mother’s cell size will halve, and go from $s$ to $s/2$ (we assume for simplicity perfect binary divisions, but this assumption can easily be relaxed). What is a “sizer” in this framework? We can define it as a model where $h_d = h_d(s)$ [442]. Equally, a timer is a model where $h_d = h_d(t)$, and an adder has $h_d = h_d(s - s_0)$. At this stage, it is only intuitive, but not formally grounded, that the scatter plots of the previous section correspond precisely to these models. This problem will be discussed in section 14.3. Note that not all the choices of hazard rates will guarantee a steady-state cell size distribution. As a particular case, one can consider a constant division rate $h_d(t) = r$, which is a simple Poisson process (see the problem above). This is a pure timer and we expect that it will not maintain a steady-state cell size distribution (the reader can verify it, e.g. by simulations).

Beyond the forward approach, we would like to recognize the trends in the data that favor one model rather than another. In particular, we can ask which model best describes the E. coli data, presented in the first section of this chapter. This question is a “reverse problem”, and is equivalent to the inference of the hazard rate $h_d$ from data (Figure 14.2). It is a very common reverse problem for the literature, used for example in the so-called “survival analysis” in clinical studies [443]. In that case, the hazard rate typically corresponds to a one-time negative outcome (death of the patient) and the process is not repeated along lineages as in the case of cell divisions. However, the mathematical ingredients are very similar.
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<figure><img src="image_127.png" /></figure>

Math box 14.A Probability distribution of (un)divided cells

This box derives the probability distribution of (un)divided cells from the hazard rate. The probability that a cell divides between time t and t + dt is the probability of not having divided so far times the probability of dividing between t and t + dt, in turn given by the product of the hazard rate and the time interval dt, F(t|s₀, α)hₐ(s(t), t, s₀, α)dt. During the same time interval, the cumulative probability of not having divided will decrease by the same amount. Hence, we can write

$$
F(t + dt|s_0, \alpha) = F(t|s_0, \alpha)[1 - h_d(s(t), t, s_0, \alpha)dt]. \quad (14.1)
$$

In the limit of $dt \to 0$ we obtain a differential equation, which governs the evolution of our system

$$
\frac{d}{dt} F(t|s_0, \alpha) = -h_d(s(t), t, s_0, \alpha) F(t|s_0, \alpha), \quad (14.2)
$$

and whose formal solution is (for $t \ge 0$)

$$
F(t|s_0, \alpha) = e^{-\int_0^t dz h_d(s(z), z)}. \quad (14.3)
$$

Since we said that the probability of a cell division event in the time interval $[t, t + dt]$ is $P(t|s_0, \alpha)dt = F(t|s_0, \alpha)h_d dt$, the corresponding probability density is

$$
P(t|s_0, \alpha) = h_d(s, t)e^{-\int_0^t dz h_d(s(z), z)} = -\frac{d}{dt}F(t|s_0, \alpha). \quad (14.4)
$$

Alternatively, the size s can be used as a coordinate, considering for $s > s_0$,

$$
F(s|s_0, \alpha) = e^{-\int_{s_0}^{s} dz h_d^*(z, t(z))}, \quad (14.5)
$$

while $F(s|s_0, \alpha) = 0$ for $s < s_0$. Here, $h_d^*(s, t(s))dx$ is the probability of cell division in the size range between $s$ and $s + ds$. The two rates are simply related by $h_d^*(s, t(s))ds = h_d(s(t), s)dt$, where $ds/dt = h_g(s) = \alpha s$ is the rate of growth.

Consequently, there are many regression methods available in the literature, which can be transferred to our case. One of the most famous is Cox regression [444]. However, most of these regression methods need an ansatz for the parameterization of the model, which might be a nuisance, as it would require some previous knowledge. Here we consider a simpler, direct inference method, which does not need any parameterization (but is effective only with a sufficient amount of data, i.e., for many cell divisions).

Suppose for simplicity we deal with a sizer. In this case, it is possible generate an estimator for the func-
tional form of $h_d(s)$ using Eq. (14.5). By inversion, we obtain

$$
h_d(s) = -\alpha s \frac{d}{ds} \log[F(s|s_0)], \quad (14.6)
$$

where *F* can easily be estimated from data, from the cumulative fraction of undivided cells at size *s* with initial size *s*₀. In our case, we can use the mean value of the growth rate ⟨*α*⟩, since we are neglecting fluctuations in growth rate.

Since we do not know whether our assumption of a sizer applies to data, we can first combine the data and
the inference to falsify the assumption [437]. In order to do this, we can further condition our histograms
in order to fix s₀. If h<sub>d</sub> depends solely on s, then the inferred function h&#771;<sub>d</sub> should not change with varying
s₀. This is indeed the case if the procedure is applied to simulated data. However, when we apply the
same procedure to the experimental data shown in the previous section, the inferred h<sub>d</sub>(s) changes if it is
inferred for different bins of birth size s₀. Hence, we conclude that our E. coli data do not behave as a sizer,
in the sense of the hazard rate. Instead, if we consider the adder ansatz for the hazard rate h<sub>d</sub>(s &minus; s₀),
and we repeat the procedure, we find that further conditioning by birth size or time from birth does not
change our inferred hazard rate [440]. Hence, we can conclude that a hazard-rate analysis of the data
supports an adder (or at least that the data cannot falsify this simple model).
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How does the inferred $h_d$ depend on size? Curiously, for any fixed $s_0$, $h_d$ increases superlinearly for small cell sizes, then reaches a maximum after which it decreases. In other words, some cells may "miss" a cell division event and keep growing until they find a better occasion to divide. This process is called "filamentation" (because the cells that miss one or more division elongate and end up looking like filaments), and is typically the consequence of stress, but also present in stress-free growth conditions. experimental observations show that E. coli forms filaments in response to DNA damage, antibiotics, host immune systems, temperature, starvation, and many other stresses. As a consequence, size plasticity may be in many cases an adaptive strategy. The quantitative division rules of filamentous E. coli cells have been studied experimentally [445], but we lack a comprehensive mathematical model.

One very robust observation of cell division statistics, in E. coli and beyond [446, 440, 447, 448], is that the distributions of size at birth, size at division, and division times measured across conditions, collapse onto the same curve when rescaled by their mean. For instance, the distributions around these values are clearly non-overlapping: the single-cell birth-size distribution in glucose $p_{glu}(s_0)$ strongly differ from the one in TSB medium $p_{TSB}(s_0)$. In particular, the typical size at birth for E. coli growing in glucose $\langle x_0 \rangle_{glu}$ is about 2/3 the size of E. coli growing in TSB $\langle s_0 \rangle_{TSB}$ and the average division time $\langle \tau_d \rangle_{TSB}$ is TSB is half the one of E. coli in glucose $\langle \tau_d \rangle_{glu}$. This appears to be valid across different environmental conditions (e.g., nutrient quality, temperature, pH, etc.). The remarkable empirical observation is that, when comparing two conditions, the rescaled distribution is universal. If we introduce the rescaled size $\tilde{s}_0 = s_0/\langle s_0 \rangle_c$, the distribution of $\tilde{s}_0$ is universal, independent of the condition. This observation applies also to size at division, added size between divisions, interdivision time, and, to a certain extent, growth rate [447].

An obvious question that follows from this observation is how the size-scaling properties of cell-size at birth constrain the mechanisms of homeostasis and the properties of stochasticity at the single-cell level. A necessary consequence of the distribution collapse is that the processes leading to single-cell heterogeneity and homeostasis must have common underlying properties across conditions. Conditions differ because they are characterized by different dimensional scales, but, phenomenologically, division control is governed by the same underlying principles (although the key molecular players may vary). The collapse of all the distributions, when the variables are rescaled by the mean has another, stronger, consequence: whatever the division control mechanism is, it depends on only two scales, a size-scale (setting the typical cell size) and a temporal scale (setting growth rate and division time).

This constraint has strict consequences on the variability of the hazard rate across conditions. In particular, it implies that the hazard rate must take the mathematical form [449]

$$h_d(s(t), s_0, t\alpha) = \alpha \tilde{h} \left( \frac{s(t)}{\langle s \rangle_c}, \frac{s_0}{\langle s \rangle_c} \right), \quad (14.7)$$

where the function $\tilde{h}(\cdot, \cdot)$ is the same across conditions. The dependency on $\alpha$ and $t$ disappears, as the scaling of division time, implies the existence of a unique time scale. Since $\tilde{h}(\cdot, \cdot)$ is by definition adimensional, it can only depend on the product $\alpha t$, which can always be re-expressed as a function of $s$ and $s_0$, as $\alpha t = \log(s(t)/s_0)$. While this is a powerful observation, as it allows to naturally connect division mechanisms across conditions, it does not provide any evidence to a particular decisional mechanism enforcing cell division, which is encoded in the function $\tilde{h}(\cdot, \cdot)$. Addressing this question needs further experimental details.

## 14.3 Cell-division control as a discrete-time linear response process

In the previous section, we have seen how the cell-division control mechanism can be mathematically defined using the hazard-rate framework. This approach uses as a fundamental ingredient the probability per unit time of cell division $h_d$, which is, a-priori, a function of many internal cellular parameters. This approach is, in some sense, very general, as it allows to characterize any complex cellular decision process. However, this generality limits the tractability and interpretability of the model. In this section, we introduce an alternative discrete-time mathematical framework which greatly simplifies the parameterization and the interpretation of a cell-division control model [450, 451], and easily maps to the empirical
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parameters discussed in Figure 14.1.

Specifically, instead of tracking the division rate at different stages of the cell-cycle, it is often convenient to model directly the cell size at birth across different generations. In this case we can, in full generality, write

$$s_0^{i+1} = f(s_0^i, \alpha, \dots) + \eta^i(s_0^i, \alpha, \dots). \quad (14.8)$$

where $s_0^i$ is the birth size of the cell at generation $i$. The function $\eta(\cdot)$ represents a random variable whose mean is equal to 0 and having, a priori, arbitrary probability distribution. The function $f(\cdot)$ described the control over cell division. Specifically, the function $f(\cdot)$ can be simply (almost tautologically) defined as the conditional average of the size at birth at generation $i + 1$ given all the variables that contribute to cell division control (the previous size at birth, the growth rate, and others),

$$f(s_0^i, \alpha, \dots) := \langle s_0^{i+1} \rangle|_{s_0^i, \alpha, \dots}. \quad (14.9)$$

The random variable $\eta(s_0^i, \alpha, \dots)$ characterizes the fluctuations around this conditionally averaged birth size.

This formulation of the process is as general as the hazard-rate formalism as it allows to express any division probability $F(s|s_0, \alpha, \dots)$. Eq. (14.8) simply isolates the contribution of the (conditional) average size at division from the deviations from this average. This separation is useful because it allows a clear interpretation of the mechanism of division control, and because the conditional average size at division is typically accessible from single-cell experiments. For instance, a timer corresponds to $f(s_0^i, \alpha) \propto s_0^i$, where the proportionality constant equals $\exp(\alpha\tau_d)/2$. A sizer corresponds to $f(\cdot)$ being a constant, independent of the initial size $s_0^i$. Along the same lines, an adder is defined as $f(s_0^i, \alpha) = (s_0^i + \Delta(\alpha))/2$, where $\Delta(\alpha)$ corresponds to the (average) added size. The formalism also shows how there is a continuum of possible intermediate behaviors besides these three limit cases.

Given the facts that growth is exponential, and the distribution of sizes at birth is approximately Lognormal [440, 447], it is once again convenient to introduce the logarithmic size $q_0^i = \log s_0^i$. One can derive the dynamics of the variable $q_0^i$ as a function of the dynamics defined in Eq. (14.8) [449]. Since the fluctuations of this variable are small, this dynamics is fully specified by a set of linear-response parameters $\lambda_{ab}$ relating the main observables (i.e. in our case each of the variables $a, b$ can be $q_0, \alpha, \tau, G$).

The linear-response framework offers a flexible and analytically tractable tool to formulate and explore different models of division control. The models can be constrained by correlation patterns measured in data, quantified for example by covariances, which relate to the coupling parameters $\lambda_{ab}$. However, the question remains of whether such models are consistent with data. For E. coli data, the linear-response framework predicts the correct consistency relations between experimental measurements, thereby confirming its usefulness to characterize empirical data [451]. A second, more biologically relevant, question is identifying the biological mechanism reproducing the observed dependency patterns. As already discussed, the observation that $\lambda_{qq} \sim 0.5$ is a strong indication of adder-like size-control mechanisms [450, 440, 449, 451]. Interestingly, one can show that the non-zero correlation between growth rate and loginitial size $\langle \delta\alpha^{i+1} \delta q_0^i \rangle$ can be explained because of the correlation between mother and daughter single cell growth rates (the presence of a non-zero value $\lambda_{\alpha\alpha}$ and a dependency of the division size on the growth rate (a non-zero term $\lambda_{q\alpha}$). Such a relation between parameters point to some dependency on the size at division on the single cell-growth rate. For E. coli, it is possible [451] to reproduce the empirical values of these coupling parameters by assuming an adder model where the added size depends exponentially on the single-cell growth rate, following the same dependency it has on the population growth rate (this behavior will be discussed in more detail below, and is sometimes termed Schaechter's Law [452]).

## 14.4 Coordination of cell division with different cell-cycle processes

In the previous sections, we learned that E. coli single-cell dynamic data reveal the adder size-control behavior, which allows bacterial cells to maintain size homeostasis. We also discussed a mathematical framework that describes how size control is achieved, and, in particular, how the key measured variables (log-
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<figure><img src="image_128.png" /><figcaption>Figure 14.3: Comparison of different cell-cycle models including chromosome sub-periods proposed in the literature for E. coli – (A) The DNA replication-segregation cycle divides of the cell cycle into sub-periods. The B-period is the period between cell birth and initiation of DNA replication; the C-period is the period needed for completing DNA replication; and the D-period is the period between the termination of DNA replication and cell division. Finally, the I-period is the period between two consecutive initiations of DNA replication, which usually spans two generations. (B) Scheme of the ‘replication-centric’ class of models in which DNA replication-segregation sets division (first column). These models usually assume that the CD and the I periods are adders (blue lines in the third and fourth column, respectively), in agreement with data (red lines in the same panels). The G-period correlation pattern is a prediction of the model in general agreement with data (yellow vs red lines in the second column). (C) Schematic for the ‘replication-agnostic’ class of models in which a process starting at cell birth drive division (first column). These models assume the G and I periods to be adders (blue lines in the second and fourth panels, respectively). The C+D period correlation pattern is a prediction of this model which does not agree with the available data (yellow vs red lines in the third panel). (D) Schematic for the ‘concurrent cycles’ class of models in which two processes compete to set division through an AND gate (first column). These models assume the I periods to be an adder (blue lines in the fourth column) and using additional parameters predict both adders in the G and C+D periods (yellow lines in the second and third column). (E) Plotting the slope of the G versus the C+D-period allows to compare the different models with data. Schematic similar Figure 4 in [453].</figcaption></figure>

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necessary to partition the cell cycle into sub-periods. We then present the key elements and observations around the debate on whether and how DNA replication and genome segregation is limiting cell division in *E. coli*. In presenting this debate we aim to (i) highlight the positive and innovative aspects of some of the cornerstone studies of recent years, (ii) provide the reader with robust tools necessary to compare mathematical models against data. Finally, we conclude the section by underlying a few open questions.

It is a classic question in biology [454, 455] how cells achieve the precise coordination of the cell cycle with chromosome replication and segregation is necessary for cell survival. DNA replication defines a way to subdivide the cell cycle into sub-periods. In *E. coli*, the period between cell division and initiation of DNA replication is normally referred as the B-period. The C-period is the period needed to complete replication. Bacterial DNA is organized in circular chromosomes which replicate starting from a well-defined "origin" region (called *ori* locus). The replication machinery moves bi-directionally, and the two "replication forks" proceed approximately at the same speed and terminate in a "terminus" region of the chromosome called *ter* locus [456, 457, 458]. For *E. coli* cells dividing at mean interdivision times from about 20 minutes to about one hour, the replication speed is approximately constant, resulting in an approximately constant C period of around 40 minutes [459]. The D-period is the period that lasts from the end of replication to the next division which thus includes segregation and septum formation. Note that the inter-division time, i.e. the time between two consecutive division events, can be as short as 20 minutes in *E. coli*. How can a cell with a division time shorter than the C-period duration have at least two copies of the DNA? Classical studies have shown that *E. coli* and other bacteria can set up multiple overlapping rounds of replication, as summarized by Cooper and Helmstetter in 1968 [455]. For example, a cell at birth is already replicating DNA and has two forks. During the cell cycle, two new initiation events take place, which will only terminate in the daughter cells [460]. We will refer to the "G-period" and the "I-period" as the periods between two consequent division and initiation events, respectively.

As briefly mentioned in the introduction of this chapter, the recent single-cell experiments allow to score initiation and termination of DNA replication by fluorescently tagging proteins involved in the formation of the replication forks or directly the *ori* locus [461, 462, 463, 464, 453]. The scoring of initiation and termination makes it possible to produce the size-growth, and the equivalent adder, plots for any of the sub-periods BCD<fnref n="2" /> as well as for the G- and I-periods (jointly). In the remainder of this section, we will refer to the slope of the size-growth plot of a sub-period X (X= B,C,D,G, or I) as $\lambda_X$, and to the slope of the corresponding adder plot as $\zeta_X$. The two slopes are linked by the equation $(1 - \lambda_X) = \frac{\zeta_X + 1}{Q_X}$, where $Q_X = \exp(\langle\text{growth during X}\rangle)$ (see Mathematical Detail Box 14.B).

Having formally defined sub-periods for the cell cycle and the corresponding linear-response formalism, we now proceed by discussing a schematic overview of the experimental observations in *E. coli* that any mathematical model should reproduce:

* The G-period shows an adder behavior, ($\lambda_G = -0.5$, $\zeta_G = 0$) [439, 440].

* The C-period duration is approximately constant across cells and experimental conditions with, a tendency to increase for slow growth rates and the C-period generally shows a timer behavior<fnref n="3" /> ($\lambda_C = 0$, $\zeta_C = Q_C - 1$) [465, 466, 467, 461, 457].

* The I-period shows an adder behavior, ($\lambda_G = -0.5$, $\zeta_G = 0$) [468, 464, 463].

* The CD-period shows an adder behavior ($\lambda_{CD} = \frac{Q_{CD}-1}{Q_C}$, $\zeta_{CD} = 0$) [464, 469].

* The single-cell growth rate and the duration of the CD period are inversely proportional [462].

Other interesting observations that are considered in the mathematical models we will present shortly are

* *E. coli* cells divide symmetrically with a narrow distribution of division length with CV = 0.05 [437]. Note that this CV is lower than the CV of both the growth-rate distribution (CV ≈ 0.1) and interdivision time distribution (CV ≈ 0.2).

* The growth rate of the mother cell is correlated positively with the growth rate of the daughter cells, with a Pearson correlation of around 0.5 [436].

<footnote n="2">²Note that under fast-growing conditions the termination is experimentally harder to score reliably and hence in many studies the C and D periods of single cells are considered together as a "CD period".</footnote>

<footnote n="3">³Given the difficulty in observing the C-period in single cells, this last question requires further experimental investigation.</footnote>
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Σ Math box 14.B Linear formalism and adder plots

This box shows how to translate the linear response ("λ-formalism") to an equivalent formalism based on the slopes of adder plots ("ζ-formalism"). The interested reader can find more information in [450, 451, 468, 470, 453]. As discussed previously, Eq. (C.2) makes it possible to estimate the linear-response parameter λ in ex- perimental data from the covariance of log-size fluctuations between subsequent generations, by noticing that

(1 − λ$_{G}$ ) = 〈 δq i 0 +1 δq i 0 〉 σ 2 $_{q$_{0}$}$, where we refer to λ in Eq. (C.2) as λ$_{G}$ , to highlight the fact that this equation refers to the G-period. Exponential growth dictates that 2 s i +1 0 = s i $_{0}$e α $^{i}$τ$^{i}$, where s i $_{0}$, α$^{i}$, and τ i are the size at birth, the growth rate and the interdivision time, respectively. For the cell cycle i one can expand the logarithmic growth G i G := α $^{i}$τ i around its average value (( G$_{G}$ ) ≃ log 2) in terms of variations around the logarithmic size at birth q i 0 := log s i $_{0}$. Following this procedure, the cell size at birth of generation i + 1 within a lineage can be expressed as a function of the parameters of generation i , as follows,

$$
2s_0^{i+1} = Q_G(s_0)^{1-\lambda_G} \langle s_0 \rangle^{\lambda_G} + \nu_0^i, \quad (14.10)
$$

where Q$_{G}$ = e ( G$_{G}$ ) = exp 〈 log s$_{d}$/s$_{0}$ 〉 , s$_{d}$ is the cell size at division and ν i 0 is a discrete-time Gaussian noise with mean zero and standard deviation σ$_{s}$$_{0}$ . Expanding around the average size, for small fluctuations we obtain a mapping between added size and slope of the size-growth plot,

$$
2s_{0}^{i+1} = Q_{G} \langle s_{0} \rangle + (1 - \lambda_{G}) Q_{G} \delta s_{0}^{i} + \nu_{0}^{i} \\
\delta \Delta_{G}^{i} = + [(1 - \lambda_{G}) Q_{G} - 1] \delta s_{0}^{i} + \nu_{0}^{i}.
$$

Here Δ i G = s i f − s i 0 is the added size during a cell cycle, and δ Δ i G = Δ i G − 〈 Δ i G 〉 is its fluctuation. Hence, by definition, the term in square brackets must be the slope of the adder plot

$$
\zeta_G := (1 - \lambda_G) Q_G - 1. \tag{14.11}
$$

Solving the equation for λ$_{G}$ , we get

$$
(1 - \lambda_G) = \frac{(\zeta_G + 1)}{Q_G}, \qquad (14.12)
$$

which can be used (assuming as usual small fluctuations) to convert the slope ζ$_{G}$ of the adder plot into the slope of the size-growth plot λ$_{G}$ , and vice-versa.

The mathematical models proposed in the literature can all be described with the general framework we provided so far. However, they are different in terms of ingredients and relevant variables (Fig. 14.3). Specifically, they can be grouped into two broad classes with fundamentally different views on the role of DNA replication, its impact on cell division control, and ultimately on how the cell division and replication cycles are coupled [470, 471, 457, 463, 464]. A class of ‘replication-centric’ models see the completion of DNA replication as the crucial checkpoint for cell-cycle progression, which fundamentally limits division and initiation events [462, 464]. Instead, ‘replication-agnostic’ models assume that cell division is limited by a cell cycle-related process such as septum or cell wall formation and not by DNA replication [472, 463].

The linear-response theory over sub-periods coupled with the new-generation experimental observations on single cells gives us a powerful tool to compare the different models (see Box 14.C). Crucially, while the slopes of the size-growth plots are ultimately correlation patterns, the interpretation of the causal link between them changes across different models. For instance, the replication-centric models generally assume that two parameters among λ$_{I}$ , λ$_{B}$ , λ$_{CD}$ are input variables, fixed by an underlying molecular mechanism, while λ$_{G}$ is an output of the model, i.e. an emergent correlation pattern predicted by the model. In contrast, the replication-agnostic models assume a mechanism for the G-period ( λ$_{G}$ is fixed), and the other correlation patterns are outputs of the model. Hence, the observed relationships between linear-response constants across conditions can be used to select a specific model. In the following, we present replication-agnostic theories first, then replication-centric models, then we introduce a class of models that find a solution of this dichotomy.

The replication-centric models are in line with the classic views on the E. coli cell cycle, but they are chal-
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<figure><img src="image_129.png" /></figure>

Math box 14.C Comparing cell-cycle sub-periods models with data

This box describes the quantitative tools necessary to systematically compare cell-cycle sub-periods models with data using the linear-response formalism and size-growth plots. Since the formalism may become very heavy, to avoid complications we will present the the case of slow-growth conditions, in which there are no overlapping replication rounds. In addition, we will assume that the growth rate is a constant parameter and we will assume perfectly symmetric division.

Replication-centric models assume $\lambda_{CD}$ and either $\lambda_B$ or $\lambda_I$ to be input parameters in the model. Here, we focus on the case in which $\lambda_{CD}^*$ and $\lambda_I^*$ are fixed, which is the case for the Cooper and Helmstetter, Ho and Amir, and Witz et al models [455, 473, 474]. In these models, one has that $\delta q_I^{i+1} = (1 - \lambda_I^*)\delta q_I^i + \alpha\nu_I^i$ and $\delta q_0^{i+1} = (1 - \lambda_{CD}^*)\delta q_0^i + \alpha\nu_{CD}^i$, where $q_0^i$ and $q_I^i$ are the logarithmic sizes at birth and initiation of the cell cycle $i$, respectively; $\alpha$ is the growth rate, and $\nu_I^i$ and $\nu_{CD}^i$ are the white noise contribution related to the I and CD periods, respectively. In this class of models, $\lambda_G$ and $\lambda_B$ are mathematically linked to $\lambda_{CD}^*$ and $\lambda_I^*$, which provides predictions that can be validated or falsified with data:

$$ (1 - \lambda_G) := \frac{\langle \delta q_0^{i+1} \delta q_0^i \rangle}{\sigma_{q_0}^2} = \frac{(1 - \lambda_{CD}^*)^2 (1 - \lambda_I^*) \sigma_{q_I}^2}{\sigma_{q_0}^2}, \qquad (14.13) $$

$$ (1 - \lambda_B) := \frac{\langle \delta q_I^i \delta q_0^i \rangle}{\sigma_{q_0}^2} = \frac{(1 - \lambda_{CD}^*) (1 - \lambda_I^*) \sigma_{q_I}^2}{\sigma_{q_0}^2}. \qquad (14.14) $$

Note that by combining (14.13) with (14.14), we also get the relationship

$$ (1 - \lambda_G) = (1 - \lambda_{CD}) (1 - \lambda_B). \qquad (14.15) $$

lenged by recent findings [455, 475, 470, 472, 465]. The 1968 Cooper and Helmstetter model was based only on the available population-average data at that time. The model posits that cell division happens within a defined period (CD) of time after initiation. Shortly after, Donachie [475] combined the Cooper and Helmstetter observation of a constant (population average) CD period with the even older observation that population-average cell size increases with the growth rate with a trend that is compatible with an exponential (Schaechter's law [452], which we mentioned above) and postulated that the population-average mass-per-origins is constant with the growth rate. Crucially, the classic paradigm by which replication limits division rested on indirect conclusions based on population averages, but these assumptions needed to be verified by single-cell data, which showed that things are much more complex [431].

In recent times, Ho and Amir [473] were the first to connect the Cooper-Helmstetter-Donachie ideas with the new observation of adder correlation patterns over the G-period. The authors assumed an adder mechanism during the I-period and a timer mechanism during the CD period. This model produces (in the limit of small noise in the timing of the CD period) an adder behavior in the G-period. Note that in this model $\lambda_I = -0.5$ and $\lambda_{CD} = 0$ are inputs while $\lambda_G \approx -0.5$ is an output of the model. This model, by definition, fails in reproducing the adder behavior in the CD period (which was not known at the time). Although it turned out to be an oversimplification, this work has the merit of connecting the old theories with new single-cell data into a simple and elegant replication-centric model.

The first studies measuring the initiation of DNA replication in single cells [461, 462] brought two new experimental pieces of evidence into the field: they observed the duration of the CD period was inversely proportional to the single-cell growth rate and that the C period does not display any size compensation. Based on their data, Wallden and coworkers proposed a replication-centric model with a sizer in the B-period ($\zeta_B = -1$), which was later falsified [468, 464, 463]. A subsequent study by a different group [464] measured consecutive initiation events in single cells and observed three adders in the G, I, and CD periods. They then designed an improved version of the Ho-Amir model (already proposed for mycobacteria [476]) in which the initiation of DNA replication triggers both the next initiation and a division event with an adder mechanism. In this model, the adder in the G-period is an output of the model, which emerges from the adder in I and CD when the growth rate is a random variable and a sufficiently skewed asymmetry in cell division is added into the model. This replicaiton-centric model is unable to capture the growth rate - CD period inverse relationship discovered by Wallden and coworkers. However, it has the merit of
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improving the Ho and Amir model accounting for both adders in I and CD and introducing a debate over
the importance of asymmetric division.

The replication-agnostic models entered the debate more recently. Based on dynamic cell-wall and cell-
geometry measurements, Harris and Theriot proposed a model in which the completion of the division
septum, and not the chromosome, was the limiting factor for cell division [471, 472]. This model proposes
a simple molecular mechanism for the adder based on three main ingredients: (i) a crucial factor involved
in setting division is produced at a rate proportional to the cell size; (ii) this factor needs to reach a thresh-
old in the number in order the cell to divide; (iii) the factor in the next generation has to be reset, with
no history dependencies on the previous cell cycle (in the case of the septum, this is natural, as a new
septum needs to be produced from zero at every cell cycle). This model structure is still the basis for dif-
ferent mechanistic models explaining the adder during the G period, but the mechanistic factor was also
proposed to be a protein [477, 478, 463]. Further evidence in favor of a replication-agnostic view came
from experiments performed by the Jun lab [463] aiming to perturb independently the adder correlation
pattern in the G-period, while maintaining intact the adder pattern over the I-period, and viceversa. The
perturbations were achieved by inducing oscillating levels of the FtsZ protein, which forms a contractile ring
structure at the future cell-division site and of the DnaA protein, responsible for the initiation of replica-
tion, respectively. The authors interpreted the results of these experiments as a proof that the replication
and division cycles are independently regulated, and in particular that completion of DNA replication and
segregation is not a limiting factor for cell division. Additionally, the authors re-interpreted the 'molecular
adder' model proposed by Harris and Teriot, suggesting that the FtsZ may be the "adder protein" setting
division. This work has the merit of providing precious experimental information. However, the model
fails to explain the adder behavior over the CD period, as well as the correlation patterns related to how
the replication and the division cycles are coordinated [468, 470, 453].

The replication-centric and replication-agnostic views have been firmly opposing each other in recent years (see e.g. [479, 474, 480]). However, a standpoint that is gaining consensus is that neither of these views is able to capture the full complexity of the correlation patterns in the data [468, 470, 453, 465, 457, 481]. The recently proposed "concurrent-cycles" scenario [468, 470, 453] bridges the two opposing views and is in better agreement with the data compared to all the above models. The key innovative element in this theoretical framework lies in the assumption that there is no unique process limiting cell division. Rather a set of competing processes have to be completed before division, and some "downstream control" module (modelled as a logic gate) has to process the input from these processes. In its original formulation [468, 470], based on the available data the competing processes are the DNA replication processes defined by an adder in the I-period, a timer in the replication-segregation period cycle, and a cell division process that adds constant size between two consecutive divisions (division-related cycle). The division is decided by an AND gate, which triggers when both of two actions are completed, the interdivision period is complete and the replication-segregation period is complete. Therefore, the AND gate selects the slowest of the two random processes (which vary across single cells) to set the timing. Note that in this framework the CD period can be set by the intrinsic replication-segregation period of this is the slowest process, or by the interdivision period in case this other process is the slowest one. The concurrent-cycles framework makes precise predictions on how the sub-periods correlations of size change when either the replication-related or the division-related cycles are perturbed. Recently, experiments in which cell wall insertion is delayed confirmed the prediction of the model [453]. Other recent studies proposed similar frameworks, adding mechanistic details, where the onset of constriction at the divisome [481] and/or a "progression control complex" including the chromosome and the divisome play the role of the gate deciding cell division [465, 457]. Technically, concurrent cycle models need an additional set of parameters compared to the replication-centric and agnostic models (see Box 14.D). These parameters are ultimately summarized by one extra relevant parameter, which can be expressed as the probability that the division-related process to sets division (in a given cell cycle). Thus, the replication-centric and replication-agnostic models can be seen as limit cases of the concurrent-cycles framework, where this probability is zero or one respectively.

Despite the large improvement that the concurrent-cycles framework provides in the agreement with data,
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processes limiting division varies under different conditions. Recent surveys of the available data [453, 481] suggest that the probability of a chromosome-agnostic cycle increases with increasing growth rate. At very slow growth (interdivision times of 300 minutes or more), it has been suggested that replication-segregation is the limiting process. Additionally, we currently do not know what tunes such probability and what the role of the growth rate may be. We also do not know how many concurrent processes there are and which precisely are the relevant players at the molecular level. Finally, the regulation of initiation of DNA replication could also be set by a "gate" integrating a set of processes, a hypothesis that remains underexplored in the literature.

<figure><img src="image_130.png" /></figure>

## Math box 14.D The concurrent-cycles framework

This box provides the mathematical relationships that correspond to the ones appearing in Box 14.C for the more general concurrent-cycles framework. Given the complexity of this model, we restrict to the case of no overlapping rounds. In particular, we will show how Eq. (14.15) is no longer valid in the concurrent-cycles framework (without the need to include additional ingredients such as asymmetric division or mother-daughter growth rate correlations).

In the concurrent-cycles model, cell division is determined by the slowest of two processes. The first process is an interdivision, (chromosome-agnostic) cycle that is concluded, for generation $i$, at a log-size $q_H^i$, which is expressed as $q_H^i = q_H^i + (1 - \lambda_H^*) (q_0^i - (q_H^i - \log 2)) + \alpha \nu_H^i$, with $\lambda_H$ size control parameter of this process. The second process is a chromosome replication-segregation cycle (replication-centric), that is concluded, for generation $i$, at a log-size $q_R^i$, which is expressed as $q_R^i = q_R^i + \delta q_I^i + \alpha \nu_I^i$. Note that this equation assumes a timer for this process, $\lambda_{CD'}^* = 0$, where $CD'$ identify the time needed for completing DNA replication, which is identical to the measurable CD-period only when this second cycle sets division. The cell size at division is determined by the slowest process, i.e. $q_d^i = \max(q_H^i, q_R^i)$. The initiation of DNA replication decides the next initiation independently on the size at birth or division, generating the fluctuation around the logarithmic size at initiation that we already found in Box 14.C, $\delta q_I^{i+1} = (1 - \lambda_I^*) \delta q_I^i + \alpha \nu_I^i$.

To calculate the fluctuations of the logarithmic size at division, we assume that the replication-centric process sets the division of generation $i$ with probability $p_H$ independently on $q_0^i$ and $q_I^i$. With this assumption, and considering $\lambda_H^*, \lambda_I^*$ and $\lambda_{CD'}^* = 0$, the model predicts the following values for the strength of the size-growth plots in the B-, CD- and G-period,

$$ (1 - \lambda_B) = (1 - \lambda_{CD}) (1 - \lambda_I) \frac{\sigma_{q_I}^2}{\sigma_{q_0}^2} \qquad (14.16) $$

$$ (1 - \lambda_{CD}) = (1 - p_H) + p_H (1 - \lambda_H^*) (1 - \lambda_B) \frac{\sigma_{q_I}^2}{\sigma_{q_0}^2}, \qquad (14.17) $$

$$ (1 - \lambda_G) = (1 - p_H) (1 - \lambda_B) + p_H (1 - \lambda_H^*) \qquad (14.18) $$

Overall, the concurrent-cycles model allows to match the experimental trends in the size-growth plots with an additional parameter ($p_H$). In particular, it allows to break the relationship in Eq. (14.15) without including asymmetric divisions or mother-daughter correlations in growth rates [468, 470, 453].

## 14.5 Protein sectors and cell division

This chapter focuses on quantitative descriptions of the cell cycle and cell division control, and it is natural to wonder whether and how these consideration relate to the topic of previous Chapters 8 and 10 which deal with resource allocation models where cell growth is set by catabolism and biosynthesis. There is a strong link between regulation of growth and cell-cycle progression, which remains a largely open area of investigation both in biology and in quantitative biology / physics of living systems. This section discusses some recent models aimed to describe some specific aspects of the coordination between cell growth and cell-cycle progression. We will start by presenting the main questions that we want to address with the aid of mathematical models. Then we will discuss the main ideas and ingredients behind the models that address these questions, and present some relevant predictions that can be tested and validated against experimental data.

The maintenance of an interplay between cell growth and cell cycle is crucial for the correct functioning of the cell. Specifically, a cell has to adapt both growth and division rates concertedly when either one is perturbed. For example, the response and adaptation to environmental stresses, such as sudden shifts
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in nutrient conditions or exposure to drugs or toxins, requires the ability to reprogram in a coordinated way cell growth and cell division. Consequently, cells across all kingdoms of life have developed specific mechanisms to precisely coordinate cell cycle progression with cell growth and biosynthesis [142, 482, 434, 483, 438, 484]. There are many mechanisms involved in this coordination, and we lack a complete and coherent quantitative understanding of how this coordination works in different contexts. Sometimes we even lack simple ways to frame questions concerning the effects on cell cycle progression of cell growth perturbations/inhibitions, or the effects of cell growth of cell-cycle perturbations (such as cell cycle arrest).

To formulate and address these questions quantitatively, we would need a theoretical framework where both growth physiology (as in "how does a cell grow?") and cell-cycle decisions/progression (as in "how does a cell decide when to divide?") aspects are allowed to play a role and influence each other. However, while both cell growth and cell cycle progression alone have been subject of intense study in the past (especially in bacteria [432], but see ref. [485] for a recent review of these themes in eukaryotes), comparatively little effort has been directed so far toward the development of such unified framework. Nonetheless, recent work has advanced our quantitative understanding of the cross-talk between cell growth and cell cycle progression in bacteria. The remainder of this section will focus on discussing these aspects.

Relatively to the bacterium *E. coli*, recent and current efforts aimed at integrating already existing coarse-grained models of cell physiology and cell cycle control. More precisely, several studies have extended the classic proteome allocation theory, (presented in chapters 8 and 10), which has proven successful in describing several physiological laws, to include also a cell-division proteome sector "X", whose dynamics should implement cell-division control (or cell-cycle progression control) strategies at a phenomenological or molecular level (Fig. 14.4). The current models for *E. coli* usually include a threshold accumulation process for cell division, i.e., proteins of the division sector accumulate during cell cycle progression up to a threshold level that triggers cell division. The previous section has mentioned some candidate molecular players for this accumulation (the FtsZ protein and the cell wall insertion).

Let us take a closer look at the ingredients of this modeling framework. The two main ingredients are (i) the standard proteome allocation theory extended to include a division sector *X*, alongside to the standard main sectors (see Chapter 8), *Q* (house-keeping), *R* (ribosomes), *P* (catabolism and transport), together with (ii) a threshold-accumulation division strategy to set the decision to divide (Fig. 14.4A). Note that the fact that the division factor *X* is a protein is an implicit assumption in these framework and experimentally things could be more complex. Crucially, the fact that cell division is a proteome sector couples the rates of cellular growth and division, by controlling the synthesis of division proteins. Specifically, the models encode a trade-off between ribosomes and division protein synthesis, which as we will see determines many salient predictions.

Box 14.E shows how these ideas and ingredients can be translated into a mathematical model. The framework that we are now going to discuss is consistent with different models recently developed in the literature [487, 488, 489, 490, 483].

In order to exemplify how this framework can generate relevant predictions, we dedicated an appendix "Growth Laws" at the end of this document where some concrete examples taken from the literature are discussed. The mathematical derivations are not exhaustive, but aimed to give the reader a feeling of the "recipe" followed to obtain a given prediction starting from the model's ingredients. The interested reader should have sufficient information to work out the mathematical calculations autonomously or follow the complete derivation in the cited references (for example by Serbanescu et al. [487, 488]).

## 14.6 Control of cell division across species and kingdoms

The concepts described in the previous sections are widely applicable, but there are many relevant species-specific aspects, so that different crucial assumptions that we have taken so far might break down for different species and kingdoms. Additionally, it should be noted that the approach described here is purely phenomenological, while a biological investigation might be concerned with the detailed molecular players responsible for the cell division and cell-cycle progression decisions. Even in this case, the approach is useful and is being applied in recent work. For example, if the goal is to understand how the size control
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Σ Math box 14.E A mathematical model

The model consists of two different layers of dynamical equations, and one relationship connecting them. The first set of equations describes cell growth and division as cellular processes

$$ \frac{ds}{dt} = \lambda s, \quad \frac{dX}{dt} = k_X s - \frac{d_X}{m_X} X, \quad X(\tau_d) = X_{th} \Rightarrow \begin{cases} s(\tau_d) \to s(\tau_d)/2 \\ X(\tau_d) \to 0 \end{cases}, \quad (14.19) $$

where cell size s (mass or volume) grows exponentially at a rate $\lambda$ ($[\lambda] = [T]^{-1}$), while division proteins X, of mass $m_X$ being synthesized and degraded at rates ($k_X$ ($[k_X] = [s]^{-1}[T]^{-1}$), $d_X$ ($[d_X] = [M][T]^{-1}$)), accumulate until a threshold amount of them is reached and cell division occurs, after that cell size is divided exactly in half and division proteins number is reset to zero.

The second set of equations describes the dynamical allocation of the proteome and the biosynthesis layer underlying cell growth, as follows

$$ \frac{dA}{dt} = \frac{1}{m_a} \left( k_n P - ak_t R f_a + \sum_{P_i \in \{Q, P, R, X\}} d_{P_i} P_i \right), \quad (14.20) \\ \frac{dP_i}{dt} = \frac{1}{m_{P_i}} (ak_t f_{P_i} R f_a - d_{P_i} P_i) . \quad P_i \in \{Q, P, R, X\}. $$

According to Eq. (14.20), free amino-acids (A) are produced from import/catalysis of nutrients at a rate $k_n$ ($[k_n] = [M][T]^{-1}$) per number of catabolic/transport proteins P, and from protein degradation, occurring at a rate $d_{P_i}P_i$ (where $d_{P_i}$ ($[d_{P_i}] = [M][T]^{-1}$) is the degradation rate) for each specific sector. Free amino-acids are taken up to synthesise each proteome sector $P_i$ at a rate equal to the number of active ribosomes ($Rf_a$), times the fraction of ribosomes synthesising the specific sector $f_{P_i}$, times an overall protein translation rate, which in this particular model is equal to a constant translation rate per ribosomes $k_t$ ($[k_t] = [s][T]^{-1}$) times the concentration of free amino-acids $a \equiv (m_a A) / ([a] = [M][s]^{-1})$.

Finally, there must be a connection between the two levels of description, in the sense that cellular rates should be regarded as the result of the underlying biosynthesis dynamics. To make this connection explicit, we write the equation

$$ s = \gamma M = \gamma(m_A A + m_P P + m_R R + m_Q Q + m_X X), \quad (14.21) $$

representing mass conservation (if "size" stands for "mass" $s = M$), or the assumption of constant density (if "size" stands for "volume" $s = V$), verified in E. coli for population averages but not for single cells, or for certain perturbations [435, 486]. Together, Eqs. (14.19), (14.20) and (14.21) fully specify the model.

(A) Unified coarse grained whole-cell model

<figure><img src="image_131.png" /></figure>

(B) Model predictions

<figure><img src="image_132.png" /><figcaption>Figure 14.4: Ingredients and predictions of modeling frameworks integrating sector models with cell-division control – (A) The framework unifies growth and cell division by extending the standard proteome allocation model to include a division sector X, implementing a threshold-accumulation process setting the decision to divide. (B) The model is naturally suited to uncover general relationships and growth laws involving proteome composition and growth rate, as well as trade-offs between different proteome sectors. The inclusion of a division protein sector X regulating cell division allows the model to make predictions on cell size control and study the transient dynamics in nutrient shifts.</figcaption></figure>