Tumorigenesis requires multiple genetic changes. Mutator mutations are mutations that increase genomic instability, and according to the mutator hypothesis, accelerate tumorigenesis by facilitating oncogenic mutations. Alternatively, repeated lineage selection and expansion without increased mutation frequency may explain observed cancer incidence. Mutator lineages also risk increased deleterious mutations, leading to extinction, thus providing another counterargument to the mutator hypothesis. Both selection and extinction involve changes in lineage fitness, which may be represented as “trajectories” through a “fitness landscape” defined by genetics and environment.Here I systematically analyze the relative efficiency of tumorigenesis with and without mutator mutations by evaluating archetypal fitness trajectories using deterministic and stochastic mathematical models. I hypothesize that tumorigenic mechanisms occur clinically in proportion to their relative efficiency. This work quantifies the relative importance of mutator pathways as a function of experimentally measurable parameters, demonstrating that mutator pathways generally enhance efficiency of tumorigenesis. An optimal mutation rate for tumor evolution is derived, and shown to differ from that for species evolution.The models address the major counterarguments to the mutator hypothesis, confirming that mutator mechanisms are generally more efficient routes to tumorigenesis than non-mutator mechanisms. Mutator mutations are more likely to occur early, and to occur when more oncogenic mutations are required to create a tumor. Mutator mutations likely occur in a minority of premalignant lesions, but these mutator premalignant lesions are disproportionately likely to develop into malignant tumors. Tumor heterogeneity due to mutator mutations may contribute to therapeutic resistance, and the degree of heterogeneity of tumors may need to be considered when therapeutic strategies are devised. The model explains and predicts important biological observations in bacterial and mouse systems, as well as clinical observations. Tumorigenesis is a multistep process somatic mutator mutations occurring as a step in the evolution of somatic cells towards malignancy.The mutator hypothesis states that mutations leading to enhanced genomic instability (termed “mutator mutations”) drive cancer pathogenesis by accelerating the acquisition of oncogenic mutations. Originally formulated around DNA polymerases and repair enzymes On the contrary, it has been argued that mutator mutations (MM) are unnecessary for cancer development, and that the observed incidence rates of cancer may be explained by mutations occurring at the normal rate in conjunction with multiple rounds of lineage expansion and selection Efficiency is defined as the expected number of malignant lineages generated up to and including a reference timepoint by any particular tumorigenic mechanism. This shifts the issue from analyzing the waiting time to a single cancer cell, and fitting it to epidemiologic data, to the evaluation of the relative efficiencies of mutator and non-mutator pathways in cancer lineage production.A novel approach was recently suggested, based on the wider perspective that all potential mechanisms of tumorigenesis are in play, but those which produce malignant lineages most efficiently are most likely to contribute to clinical cancers In this framework, mutator mutations, lineage expansion, and selection are not mutually exclusive and could all simultaneously contribute to tumorigenesis. It is also noted that the conversion rate of normal cells to cancer cells (“cancer lineage birth rate”) likely far outstrips the number of clinically observed cancers, due to numerous malignant and premalignant lineages being eliminated by immune surveillance, failure to establish a blood supply, or competition from other premalignant lineages. Thus models which match the cancer lineage birth rate to clinical cancer incidence may have inherent limitations. It may be more relevant to evaluate the potential contribution of mutator mutations to the efficiency of tumorigenesis, as opposed to whether mutator mutations are necessary to explain a rate of cancer lineage birth rate equal to that of clinically observed cancer incidence. Furthermore, attempts to compare absolute theoretical cancer rates to absolute observed cancer rates are very sensitive to the underlying parameters and other assumptions, leading to variability in conclusions An analysis of the relative efficiency of tumorigenesis with and without a somatic mutator mutation, in the absence of lineage expansion (LE), demonstrated that mutator mutations enhance tumorigenic efficiency under many realistic scenarios, despite the need for an extra mutation step to acquire the mutator mutation itself Mutator lineages are also more likely to suffer deleterious mutations that reduce their fitness and potentially lead to extinction. This effect has been termed negative clonal selection (NCS) fitness landscape, or the multidimensional space representing cellular fitness, as a function of cellular genetic makeup within an environmental context. Pathways through this fitness landscape are termed fitness trajectories. Trajectories of special interest for tumorigenesis are those which begin with a normal cell and end with a transformed malignant cell.In order to account for the effects of selection and expansion of fitter lineages, as well as negative clonal selection, I systematically consider the This paper presents mathematical models which represent the general case of tumorigenesis across a variety of fitness trajectories, including multiple situations where the mutator lineage suffers reduced fitness (NCS), or achieves increased fitness leading to lineage expansion (LE). Four cases (“fitness trajectories”), which differ in the fitness of intermediate lineages in the tumorigenic process , are conincremental lineage expansion case, the lineages acquire a fixed increment in fitness with each successive oncogenic mutation, finally achieving their maximum fitness when they have acquired a full complement of oncogenic mutations. Given that most of the increased tumorigenic efficiency due to a mutator mutation can be captured by the case in which the mutator mutation is an initial step cooperative lineage expansion case with early mutator mutation, there is no increase in fitness until a subset of oncogenic mutations have been acquired, at which point the fitness increases rapidly and lineage expansion begins. Additional oncogenic mutations may then be required to achieve the fully malignant phenotype. The mutator mutation occurs early, e.g. at any point before the lineage expansion. In case 3, the cooperative lineage expansion case with late mutator mutation, the situation is analogous to case 2 except that the mutator mutation occurs after the onset of lineage expansion, during the period when additional oncogenic mutations are occurring towards reaching a fully malignant phenotype. Since cases 2 and 3 are alternate mutually exclusive subsets of the same fitness trajectory, their relative efficiencies (compared to non-mutator pathways) are additive. In case 4, the mutator and wild type lineages are subject to negative clonal selection may lead to reduction in fitness, depending on the genetic and environmental context. When a reduced fitness locus is mutated, the lineage is at risk for fitness reduction. Lineages with fitness reduction become extinct, thus potentially limiting the advantage conferred by a mutator mutation.In case 1, the While cases can be proposed that are mixtures of these four cases, it should be possible to infer their properties once these four archetypal fitness trajectories are analyzed. Thus, based on analysis of these pathways in combination  the minimum fold increase in mutation rate required from a mutator mutation before the mutator pathway has a relative tumorigenic efficiency greater than or equal to 1, termed α50%.The results for the four cases below are presented in terms of two key model outputs: (1) relative tumorigenic efficiency of mutator vs. non-mutator pathways, Nrel is the ratio of malignant lineages produced by mutator and non-mutator pathways under the specified conditions. The fraction of clinical cancers arising by mutator pathways is given by Nrel/(1+Nrel), and mutator mechanisms predominate if Nrel>1. In non-lineage expansion models, in which progression to a malignant lineage is a rare event, Nrel can be expressed as a ratio of probabilities, PrelN50%, at which mutator pathways are expected to contribute to half of clinical cancers, and above which mutator pathways predominate. The fraction of total cancers caused by a mutator pathway with a given α when compared to an alternative non-mutator pathway is given by αC/(αC+α50%C), where C is the number of oncogenic mutations required for malignant transformation. Thus, a mutator mutation must confer a minimum level of genetic instability to be relevant. In evaluating the importance of mutator pathways in a particular model, we need to determine if α50% is within a range commonly seen in known mutator mutations.α is the multiplicative factor by which a mutator mutation increases the somatic mutation rate per cell generation, e.g. the magnitude of the genetic instability. In the lineage expansion (cases 1–3) and constant fitness 50% is at or below commonly observed values of α (ca. 10–500).Mutations in base selection and proofreading generally increase mutation rates 10–100 fold optimal, which maximizes the importance of mutator pathways. This corresponds to an optimal mutation rate, kmut-optimal.For the negative clonal selection model (case 4), an additional key output is an optimal value of the fold increase in mutation rate, αR); 0≤Rp≤R, the component of R which is due to enhanced proliferation (the remainder would be due to decreased apoptosis); T, the time (in cell generations) to malignancy; NML, the number of genomic “mutator loci”, in nucleotides, mutation of which leads to genetic instability; and kmut, the mutation rate per nucleotide base per cell generation in wild type cells.The results depend on the properties of the tumor under consideration, which in turn define the inputs to the models. The key input parameters for all the models are: C, the number of oncogenic mutations required for transformation to the malignant phenotype; R≥0, the natural logarithm of the relative fitness of a malignant cell compared to wild type , an additional input parameter is introduced: D, the number of oncogenic mutations required for an increase in fitness. In the negative clonal selection model (case 4), I introduce the input parameter NRFLN-D (Nreduced fitness loci net-dominant), which is an indicator of the vulnerability of the genome to mutations which may reduce cellular fitness In the Key input parameters and the ranges over which they have been varied in the calculations, as well as key model outputs, are summarized in ML is 100, a very conservative assumption ML = 1000, α50% would decrease by a factor of 101/C (relative to the same case with NML = 100), further enhancing the potential role of mutator pathways.In the calculations, I assume NUsing the equations in R), and each successive oncogenic mutation leads to an incremental increase in fitness R/C , but for higher values of C, α50% continues to be well within commonly observed ranges. For example, when three oncogenic mutations are required for cancer (C = 3), the wild type mutation rate is low (kmut = 10−11), and the relative fitness advantage eR of malignant cells relative to wild type is 2, a 770-fold increase in the mutation rate would be required for mutator pathways to be observed in 50% of the cancers  and to some extent at C = 3 with a low wild type mutation rate /[(C+1)C] compared to the constant fitness case. However, α50% would need to increase by a factor of only {RT (C−1)/[(C+1)C]}1/C to compensate for this. For example, with the relative fitness of malignant cells eR = 2, the number of cell generations T = 5000, and the number of required oncogenic mutations C = 6, the relative tumorigenic efficiency Nrel 1∶0 is reduced over 400-fold relative to the constant fitness case. But only a 2.7 fold increase in α can restore the same relative importance of mutator pathways under these circumstances.When judged by relative efficiency Nrel 1∶0 increases with increasing wild type mutation rate kmut and increasing fold-increase in mutation rate, α, similar to the constant fitness case The analytical model , but for higher values of C, α50% continues to be well within commonly observed ranges. For example, when three oncogenic mutations are required for cancer (C = 3), the relative fitness of malignant cells eR = 2, and the wild type mutation rate is low (kmut = 10−11), an 880-fold increase in the mutation rate would be required for mutator pathways to be observed in 50% of the cancers  and to some extent at C = 3 with a low wild type mutation rate  shows that the relative contribution of mutator pathways is independent of number of cell generations T. Finally, the relative contribution of mutator pathways is shown by the analytic model (equations [13] and [16]) to be independent of the number of oncogenic mutations required for an increase in fitness, D, in contrast to the cooperative lineage expansion with ratio Nrel of malignant cell lineages produced by mutator and non-mutator pathways will be nearly equivalent to the probability ratio Prel previously derived for the constant fitness case rel (if the increased fitness includes more rapid proliferation). α50% is calculated using equation [20], and Nrel by equation [21], in In this circumstance, D oncogenic mutations occur leading to a sudden cooperative increase in fitness. After this occurs, an additional C–D oncogenic mutations must occur to complete the transformation to a malignant lineage. During this latter period, a somatic mutator mutation may occur . For bot50%. In the case of C–D = 3, for example, α50% ranges from 12 to 252, depending on various parameter values . In the cooperative lineage expansion case with late mutator mutation, a greater fitness advantage increases the pool of cells which may acquire a late mutator mutation. For example, when the number of oncogenic mutations required for cancer C = 6, the number of oncogenic mutations required for the cooperative fitness increase D = 2, the number of cell generations to cancer T = 5,000. and the wild type mutation rate kmut = 10−11, α50% is 29 when the relative fitness advantage eR of malignant cells relative to wild type is 1.2, 27 when the relative fitness advantage is 2, and 22 when the relative advantage is 7.4  and NRFLN-D (Nreduced fitness loci net-dominant), an indicator of the vulnerability of the genome to mutation rel  is calculated using equations [28–30] in The instantaneous risk of fitness reduction is the product of the mutation rate per nucleotide base per cell generation krel of mutator to non-mutator pathways does not continue to increase with greater fold increases α in the mutation rate. Increased mutation rates speed the acquisition of oncogenic mutations, but at the same time increase the risk of fitness reduction and extinction. In this type of fitness landscape, the relative efficiency Nrel of mutator compared to non-mutator pathways increases with greater fold increases α in mutation rate, until an optimum at which the growth of the malignant lineage begins to be limited by negative clonal selection. An approximate optimum for mutation rate kmut can be estimated for this circumstance from the theoretical treatment:In contrast to the other cases, the relative efficiency NNote that the treatment focuses on dominant reduced fitness mutations only. Recessive reduced fitness mutations, requiring mutation of both alleles, were found to be quantitatively insignificant mut optimal, varies from 2.1×10−10 to 3.6×10−6 per nucleotide base per cell generation. This optimum is generally higher than estimated mutation rates of wild type embryonic stem cells or somatic cells −11 to 10−9.Within the parameter ranges considered within this paper, the optimal mutation rate for tumor evolution, koptimal = kmut optimal/kmut. In the presence of an anti-apoptotic mutation RFLN-D would be reduced to a very low value, further raising the optimal mutation rate and diminishing the potential effect of negative clonal selection. Thus, mutator pathways with α>αoptimal would be more efficient if they occurred after an anti-apoptotic mutation.An approximately optimal value of the fold increase α in mutation rate due to a mutator mutation is therefore given by αrel 1:0)>0] varies depending on the strength of negative clonal selection, as shown in mut = 10−11, and a number of cell generations to cancer T = 5,000. At maximal negative clonal selection, there must be at least 5 oncogenic mutations required for malignant transformation before mutator pathways are favored. Additional more detailed results are given in Supplementary The relative importance of mutator pathways increases with increasing number of required oncogenic mutations for malignant transformation, but in contrast to the other cases, the minimal number of oncogenic mutations at which mutator pathways are favored .In particular, malignant initiation events occur from lineages with C−1 oncogenic mutations, as they acquire their Cth oncogenic mutation: The total number of malignant initiation events by time T is simply the integral of this instantaneous initiation rate from 0 to T: In summary, the instantaneous initiation rate of lineages with one oncogenic mutation is given by a first order differential equation. Based on the instantaneous initiation rate of lineages with one oncogenic mutation, and their lineage expansion, one can derive an expression for the number of cells with one oncogenic mutation as a function of time. The instantaneous rate of initiation of lineages with two oncogenic mutations is proportional to the number of cells with one oncogenic mutation at any given time. In turn, based on the instantaneous initiation rate of lineages with two oncogenic mutations, and their lineage expansion, we derive an expression for the number of cells with two oncogenic mutations at any point in time. The instantaneous initiation rate of cells with 3 oncogenic mutations is then proportional to the number of cells with 2 oncogenic mutations, and so on.Expressions for the number of cells and lineage initiations were derived for several values of n (number of oncogenic mutations) and C , and based on the algebraic details, general expressions were hypothesized for all n and C, verified for the cases explicitly derived, and proven for all n and C by mathematical induction.0,C-mut(t), is given by the product of the initial number of cells N0, the number of mutator loci NML, and the mutation rate: For mutator pathways, the mutator mutation is assumed to occur first, based on previous work suggesting that mutator pathways are more efficient if the mutator mutation occurs early Once a lineage with a mutator mutation is formed, analysis of its progress parallels that of the non-mutator pathway, except for the increase in the mutation rate constant by the factor α.50%, the minimum fold increase in mutation rate at which mutator pathways account for 50% of observed cancers (derived by setting Nrel = 1), is approximated by considering only the most rapidly growing exponential: The numbers of malignant cell initiations by both mutator and non-mutator pathways is the sum of exponentials representing lineages with 0 to C−1 oncogenic mutations. αThe exact expression, considering all exponentials, is: A and B are proportional to the number of initiation events by time T for non-mutator and mutator pathways (with α = 1), respectively. The approximate expression rel, the relative efficiency of mutator compared to non-mutator pathways, is given by the following expression when the mutator mutations increase the mutation rate by a factor of α: NIn this case, there is no change in fitness until D≤C oncogenic mutations have occurred. These first D steps, with or without a mutator mutation, can therefore be analyzed using the strategy previously outlined for the constant fitness case R per wild type cell generation (see equations [2–3] above). During this period, the fitness is also constant, although greater than it was prior to the D oncogenic mutations. The total number of cells with C−1 oncogenic mutations (by non-mutator or mutator pathways) at time T is given by the integral over t from 0 to T of the product of: the number of cells ND−1,C(t) or ND−1,C-mut(t) with D−1 oncogenic mutations  at time t; the instantaneous mutation rate per locus for conversion to cells with D oncogenic mutations ; the number of unmutated oncogenic loci at the time of conversion to cells with D oncogenic loci (NOL−D+1), the exponential lineage expansion of cells with D oncogenic mutations from time t to time T (eR(T-t)); and the probability that any progeny of this expanded lineage would have acquired the final C−D−1 oncogenic mutations in time T−t  or PC-1|D,C-mut(T−t)) for non-mutators and mutators, respectively, adjusted for the absence or presence of mutator mutations and the increased number of cell generations per unit time, if the fitness increase includes an increase in proliferation rate: Once D oncogenic mutations have occurred, the lineage has a natural log fitness advantage R, and expands by a factor of eOL-D remaining unmutated oncogenic loci, mut(T−t) for non-mutators, α kmut(T−t) for mutators, adjusted by a factor of The probability of the expanded lineage acquiring C−D−1 oncogenic mutations in time T-t is the product of the number of ways of selecting C−D−1 oncogenic mutations from Nrel, the relative number of clinical cancers due to mutator compared to non-mutator pathways at time T or earlier, is given as before by the ratio of total number of malignant initiation events at or before time T for mutator divided by non-mutator pathways (see equation [4]). α50% is again derived by setting Nrel = 1.As in case 1, we use the number of cells with C−1 oncogenic mutations to calculate the instantaneous rate of malignant initiation events at any time, integrating that from 0 to T to obtain the number of malignant initiation events at or prior to T. N50%, in the limit of significant lineage expansion, and increasingly accurate as the fitness advantage R and the fold mutation rate increase α get larger, is: rel is again given by An approximate expression for αIn this case, there is no change in fitness and no mutator mutation until D≤C oncogenic mutations have occurred. These first D steps can therefore be analyzed using the strategy previously outlined for the non-mutator pathway in the constant fitness case R per wild type cell generation. During this time, an additional C−D oncogenic mutations will also occur, with a mutator mutation occurring anywhere from the 1st to (C−D)th step in the process. During this period, the fitness is also constant, although greater than it was prior to the D oncogenic mutations.Once D oncogenic mutations have occurred, the lineage has a natural log fitness advantage R, and expands by a factor of eC−1,C-mut(T), is given by the sum, over the possible steps k at which a mutator mutation can occur, of double integrals. These double integrals from t equals 0 to T are of the product of the number of cells with D−1 oncogenic mutations at time t, ND−1,C-mut(t); the instantaneous rate of conversion of these cells to cells with D oncogenic mutations, (NOL−D+1) kmutdt; the lineage expansion from time t to T, eR(T−t); and an internal integral representing the likelihood of subsequent acquisition of the remaining C−D−1 oncogenic mutations and a mutator mutation in time T-t. This internal integral is over t' equals 0 to T-t, and the integrand is the product of the probability of having k−1 additional oncogenic mutations between time t and time t+t' before the mutator mutation occurs, Pk−1,t'; the instantaneous rate of occurrence of the mutator mutation (adjusted to account for the reduced cell generation time in the presence of a hyperproliferative mutation) at time t', eRp NML kmut dt'; and the probability that the remaining C−D−k oncogenic mutations will occur in the remaining T−t−t' cell generations, PC−1,k−1,t',t.The total number of cells with C−1 oncogenic mutations at time T by a mutator mechanism, NOL-D unmutated oncogenes, wild type cell generation by a factor In analogy with previous arguments, The number of cells with C−1 oncogenic mutations at time T by non-mutator pathways is the same as in case 2.rel, the relative number of clinical cancers due to mutator compared to non-mutator pathways at time T or earlier, is again given by the ratio of total number of malignant initiation events at or before time T for mutator divided by non-mutator pathways (see equation [4]). α50% is again derived by setting Nrel = 1.As in cases 1 and 2, we use the number of cells with C−1 oncogenic mutations to calculate the instantaneous rate of malignant initiation events at any time, integrating that from 0 to T to obtain the number of initiation events at or prior to T. N50%, in the limit of significant lineage expansion, and increasingly accurate as the fitness advantage R and the fold mutation rate increase α get larger, is: rel is given by: An approximate expression for αrel. A late mutator (case 3) is more efficient (and therefore more likely) than an early mutator (case 2) if and only if: In the case of cooperative lineage expansion, we can determine whether an early or late mutator mutation is more efficient by comparing the respective values of NFor α≫1, We see that early mutator mutations (i.e. before the fitness increase) are much more likely in the cooperative case for larger values of D.RFLN-D. Lineages with reduced fitness become extinct. The loss of fitness is described by a first order differential equation, leading to exponential decay of surviving lineages , with exponent given by minus the product of the susceptibility constant β, the number 2 (given diploid cells), α (for mutator lineages only) and the number of cell generations T. In turn, β is the product of the mutation rate constant kmut and the net number of dominant RF lociIn this case, lineages are subject to negative clonal selection (NCS), or random dominant reduced fitness (RF) mutations that are deleterious with a certain probability proportional to NRFLN-DNAs this case does not involve lineage expansion, the model can be expressed in terms of probabilities rather than expectation values.cancer, 0, NCS, is the product of the probability of surviving negative clonal selection for T cell generations (equation [24a]) and the probability of having C oncogenic mutations at time T The probability of a malignant lineage initiation by a non-mutator pathway by time T, Pmut (bearing in mind equation [24c] for β), and setting this derivative equal to zero, leading to equation RFLN-D rather than the full genome length as the size of the target which can mutate to reduced fitness, and the factor of 2 to account for a diploid genome.The maximum probability of malignancy as a function of underlying mutation rate can be found by differentiating this expression with respect to the mutation rate kcancer, 1, NCS, is the integral from t equals 0 to T of the product of the instantaneous rate of occurrence of the mutator mutation at time t (NML kmut dt); the probability of surviving negative clonal selection until time t without a mutator mutation, P0, t, NCS; and the probability of acquiring C oncogenic mutations while surviving negative clonal selection in time T–t after enhancement of mutation rate by an mutator mutation, PC-mut, 0, t, NCS: The mutator pathway probability of carcinogenesis is evaluated assuming the mutator mutation occurs first. Loss of lineages due to NCS is more rapid after a mutator mutation, but so is the acquisition of oncogenic mutations. The probability of malignant lineage initiation with a mutator mutation as step 1, P0, t, NCS, is given by [24a] and [24c] with T = t. The probability of acquiring C oncogenic mutations while surviving negative clonal selection in time T–t after enhancement of mutation rate by a mutator mutation, PC-mut, 0, t, NCS, is given by the product of: the probability of surviving negative clonal selection for T-t cell generations given a mutator mutation, e−2αβ(T−t); the number of ways to choose C oncogenes from a set of NOL oncogenes, mut(T−t)]C:The probability of surviving negative clonal selection until time t, Prel or Prel) of a mutator pathway with an initial mutator mutation to that of a non-mutator pathway in the presence of negative clonal selection is the ratio of the malignant initiation probabilities for mutator vs. non-mutator pathways, and is given by:The relative efficiency or probability (NFor (α−1)βT≪1, an alternative expression for Z must be used to maintain adequate computational precision: To reproduce the derivations above, the following identities involving factorials are required. Proofs of these identities are available on request.Table S150%: incremental and cooperative lineage expansion cases (cases 1–3). α50%, the minimum fold increase in mutation rate at which mutator pathways account for 50% of observed cancers; C, the number of oncogenic mutations required for commitment to cancer; D, the number of oncogenic mutations required for fitness increase in the cooperative lineage expansion model; eR, the relative fitness advantage per cell generation of malignant lineages compared to wild type; RP, the component of R due to enhanced proliferation rate; kmut, the wild type mutation rate per nucleotide per cell generation; T, the number of wild type cell generations at which the efficiency comparison is made (lineages with hyperproliferative mutations may have undergone more generations); LE, lineage expansion; MM, mutator mutation. For the cooperative lineage expansion case, α50% is determined by the smaller of the two values (early and late mutator mutation). For the cooperative lineage expansion case with late mutator mutation, D and RP are required for the calculation, and we assume in these examples D = 2 and RP≈0.5 R. Calculated using equations [11], [12a–h], ML, the number of “mutator loci” in nucleotides, mutation of which may lead to genetic instability, is 100. The fraction of cancers arising with an initial mutator mutation causing fold mutation increase α in their pathogenesis is given by αC/(αC+α50%C).α(0.11 MB DOC)Click here for additional data file.Table S250% for high fitness advantage R (R = 2): incremental and cooperative lineage expansion cases (cases 1–3). α50%, the minimum fold increase in mutation rate at which mutator pathways account for 50% of observed cancers; C, the number of oncogenic mutations required for commitment to cancer; D, the number of oncogenic mutations required for fitness increase in the cooperative lineage expansion model; eR, the relative fitness advantage per cell generation of malignant lineages compared to wild type; RP, the component of R due to enhanced proliferation rate; kmut, the wild type mutation rate per nucleotide per cell generation; T, the number of wild type cell generations at which the efficiency comparison is made (lineages with hyperproliferative mutations may have undergone more generations); LE, lineage expansion; MM, mutator mutation. For the cooperative lineage expansion case, α50% is determined by the smaller of the two values (early and late mutator mutation). For the cooperative lineage expansion case with late mutator mutation, D and RP are required for the calculation, and we assume in these examples D = 2 except for when C = 2 (in which case we assume D = 1) and RP = 1.31 . Calculated using [11–12a–h], ML, the number of “mutator loci” in nucleotides, mutation of which may lead to genetic instability, is 100. The fraction of cancers arising with an initial mutator mutation causing fold mutation increase α in their pathogenesis is given by αC/(αC+α50%C).α(0.10 MB DOC)Click here for additional data file.Table S3rel 1∶0, NCS, Relative Efficiency of Mutator Pathways During Negative Clonal Selection (Case 4). Prel 1∶0, NCS, the ratio of cancers arising with and without an initial mutator mutation in their pathogenesis in the presence of negative clonal selection (NCS); C, the number of oncogenic mutations required for commitment to cancer; NRFLN-D, the net number of dominant reduced fitness loci; α, the fold increase in mutation rate due to a mutator mutation; T, the number of wild type cell generations at which the efficiency comparison is made (lineages with hyperproliferative mutations may have undergone more generations). Calculated using equations ML, the number of “mutator loci” in nucleotides, mutation of which may lead to genetic instability, is 100, and kmut, the wild type mutation rate per base per cell generation, is 10−11. The fraction of cancers arising with an initial mutator mutation in their pathogenesis is given by Prel 1∶0, NCS/.P(0.07 MB DOC)Click here for additional data file.


The yeast pheromone response pathway is a canonical three-step mitogen activated protein kinase (MAPK) cascade which requires a scaffold protein for proper signal transduction. Recent experimental studies into the role the scaffold plays in modulating the character of the transduced signal, show that the presence of the scaffold increases the biphasic nature of the signal response. This runs contrary to prior theoretical investigations into how scaffolds function. We describe a mathematical model of the yeast MAPK cascade specifically designed to capture the experimental conditions and results of these empirical studies. We demonstrate how the system can exhibit either graded or ultrasensitive (biphasic) response dynamics based on the binding kinetics of enzymes to the scaffold. At the basis of our theory is an analytical result that weak interactions make the response biphasic while tight interactions lead to a graded response. We then show via an analysis of the kinetic binding rate constants how the results of experimental manipulations, modeled as changes to certain of these binding constants, lead to predictions of pathway output consistent with experimental observations. We demonstrate how the results of these experimental manipulations are consistent within the framework of our theoretical treatment of this scaffold-dependent MAPK cascades, and how future efforts in this style of systems biology can be used to interpret the results of other signal transduction observations. The yeast pheromone response system is one of the first signal transduction systems to be identified and studied in detail In the yeast system, G protein activation leads to the activation of a MAPKKK, Ste11. Ste11 activates the MAPKK Ste7, which has two possible target MAPKs, Fus3 and Kss1 Scaffolds such as Ste5 have been a subject of extensive theoretical and empirical investigations, much of the work focusing on how the scaffold controls the output response of its pathway et alRecently, several studies have shown that the yeast Ste5 scaffold plays an important role in modulating the ultrasensitivity of the Fus3 response to pheromone. These reports have shown that the scaffold-dependent Fus3 response is ultrasensitive, whereas the scaffold-independent response of Kss1 is graded To summarize, past theoretical investigations into scaffold-free MAPK systems demonstrate a tendency towards a biphasic response, while systems involving a scaffold show in theory a strong and robust graded response, which is contrary to the recent experimental findings. To address this discrepancy between theory and experimental results, we devise a new model of the yeast pheromone response system. This model is tailored to several experimental results from the recent literature, and we demonstrate how the apparent conflict can be resolved by examining how the kinetic binding parameters influence the signal response output.This work is organized as follows. First, we revisit the classic MAPK cascade in the absence of scaffold and make a detailed investigation of its output signal characteristics. Then, we devise a simplified model of the yeast scaffold-MAPK system and study its dynamics and output response as functions of the kinetic binding parameters of MAPK enzymes to the scaffold. We show how these results are consistent with both prior theoretical studies and current experimental evidence. Finally, we offer several hypotheses, based on our results, which explain how experimental perturbations to this pathway reported in the literature lead to non-obvious changes to the biphasic nature of the transduced signal.We begin by seeking a better understanding of how the MAPK system transmits its signals in response to a stimulus. We thus start with a simple implementation of the classic MAPKKKThe system consists of three enzymes, each of which has one fully activated state and one or more inactive states . We assuThe parameters for this system may be found in We find that model (1) cannot be easily adapted to include the presence of the scaffold, as the number of intermediate steps and complexes in fully active scaffold formation process is quite sizable, even though many of those steps are never detected experimentally. A full system would include a combinatorial set of partially assembled scaffold complexes, and contain both fast-paced and slow reactions. Such complexity makes the description unnecessarily opaque. Instead, we will focus on a relatively simpler system in which we assume equilibrium levels of scaffold complexes have formed, and that the no-signal resting state for the system involves all three enzymes bound to the scaffold, although not in an orientation which is immediately permissive of processive phosphorylation A signalling event by pheromone through its receptor activates two separate processes, both of which are necessary for proper signal transduction see  [1]. FirIn a scaffold-free MAPK cascade described by system (1), the rate-limiting steps are the phosphorylation events. In the presence of a scaffold we hypothesize that the rate-limiting steps are the association events of proteins with the scaffold in an orientation which allows for processive phosphorylation, in accordance with other treatments of scaffold systems. Once the elements of the MAPK cascade are attached to the scaffold, the downstream activation can happen on a faster time-scale. The pertinent reactions are (i) signal-dependent scaffold complex binding to its partner in the plasma membrane, leading to a conformational change in the scaffold bringing the enzymes into alignment with one another; and, (ii) signal-dependent enzymatic activation of the first kinase on the scaffold. The forward and reverse rates for the first reaction are Inactivation of the scaffold is a two-step processes. We consider the kinase(s) upstream of the conformational shift are inactivated as a group, and the kinase(s) downstream of the shift are as well. A scaffold with inactive Ste11 but active Fus3 is still considered an active signaling complex. Dephosphorylation of Fus3 on a scaffold which is both properly aligned and has active Ste11 is disregarded, as we assume the Fus3 will immediately be rephosphorylated. However, once a Fus3-active scaffold detaches from its membrane target  it will remain an active signaler until Fus3 is dephosphorylated, independent of whether or not the upstream enzymes are active or not.Our model is graphically presented in Finally, we consider two control parameters which guide the flow of signal through the model. The first parameter, With the state variables given in Parameters values are listed in While our model takes as its conceptual basis the hypothesis of selective activation of the scaffold, We will use model (1) to study signal response sensitivity in the MAPK signaling pathway. The sensitivity is typically measured by fitting the output response of a signaling pathway as a function of the input signal strength  to a standard Hill function:We can analyze the behavior of model (1) by making several simplifying assumptions on the Michaelis-Menton (MM) activation terms. This simplification will allow us to calculate the steady state concentrations of the output product, The classic MM equation describes how the rate of formation of product Firstly, when Secondly, if we assume that The first assumption (weak interactions) can be easy to justify over the course of a reaction: if no new substrate is created, and if the binding constant has a much higher value than the initial substrate levels, then the binding constant will always be much greater than the substrate levels. On the other hand, the second assumption (tight interactions) is more drastic and can likely be violated during the reaction duration. Nonetheless, these two simplifications provide useful analytic bounds with which to describe the nature of the system output. We term the parameter regime which favors outputs of the nature of equation 5 to be the ‘weak’ regime, and that which favors equation 8 to be the ‘tight’ regime.To study signal-response sensitivity further, we simulated model (1) for many different randomly selected values of certain parameters. We pick coefficients As seen in the left panel of We next take a small sample of these parameter sets and determine the relationship between This result is very interesting in light of what is known about the yeast pheromone pathway. The MAPKK-MAPK interaction is reported to be quite tight We perform a similar numerical experiment, over a wide range of possible binding constant values, on the with-scaffold system (model 2). We randomly calculate these constants by drawing them from a uniform log distribution  spanning from As seen in et alThe numerical surveys of both the with-scaffold  and withet al, demonstrates a significantly different response profile than the yeast system An important distinction between the mammalian and yeast scaffold-MAPK systems is the requirement, in yeast, for membrane recruitment of the scaffold prior to full scaffold activation. Mammalian scaffolds, such as the MP1 protein, do not have such a requirement for proper activation With the results from the mammalian system in mind, we next explore what role the selective activation hypothesis plays in shaping the response curve. We modify the model by allowing scaffold realignment, the et al, which showed that loss of Fus3-Ste5 association converted the Fus3 response to a more Kss1-like response.This result is a strong theoretical indicator for the selective activation hypothesis. The one extra step involved in selective activation, the recruitment of the scaffold to the plasma membrane prior to formation of signal-competent scaffold, is critically involved in the formation of the ultrasensitive response signal. In light of this result, we can now begin to analyze and understand the both studies of Takahashi and Pryciak, in which signal is artificially induced at the Ste11 MAPKKK level while bypassing the receptor; and the studies of Hao The paper by Takahashi and Pryciak employs a Fus3-driven reporter gene to measure pathway activation We can use model (2) to model this experimental system and gain understanding of how membrane binding affects ultrasensitivity. By varying the complex-membrane association rate From these simulations, we observe that a scaffold with enhanced membrane binding ability signals with a graded response to input stimulus, whereas a diffusible scaffold signals with an ultrasensitive response. This mirrors the result from Takahashi and Pryciak, if we assume that the natural effect of tethering all the scaffold complexes to the plasma membrane is to increase scaffold-signal association rate, by forcing the two components into much closer long-term proximity.We have already observed that loss of the requirement for signal-induced membrane binding prior to full scaffold activation leads to a decrease in the ultrasensitive nature of the signal response. We now look to determine whether it is the scaffold alignment or initial enzymatic activation of Ste11 which is responsible for the majority of the ultrasensitive behavior. We perform a numerical experiment in which any scaffold, whether bound to the membrane or free in the cytoplasm, can undergo the enzymatic activation step. The relative level of cytosolic scaffold phosphorylation is controlled by the parameter Thus, in this system it is clear that the alignment of Ste11, Ste7 and Fus3 on the scaffold is the critical step needed to promote an ultrasensitive response.Next, we consider a second experiment by Takahashi and Pryciak et al, showed a strong tendency for a graded response We again note here that the results of this experiment contradict current theoretical understanding of how the MAPK pathway functions in the presence and absence of scaffolds. A pathway with a scaffold based on the mammalian MAPK system, as modeled by Levchenko In this system, there is no requirement for extracellular pheromone, so we assume the scaffold remains in its locked, non-permissive configuration (cis (from a scaffold's ste11 to its ste7 to its fus3), or in trans  by one scaffold acting as an enzyme transmitting the signal to another scaffold, which serves as the substrate.Our results suggest the following hypothesis. Scaffolds in the cytoplasm will assemble into their resting state, a noncompetent form; however, now they have dominantly active ste11 attached instead of wild-type, signal-dependent ste11. There are two possible modes of full activation: either the scaffolds transmit the signal in cis transmission is not responsible for the marked increase in Hill coefficient. Consider the scenario in which a misaligned scaffold can still transmit signal from Ste7 to Fus3 at some basal leakage rate. Then, based on the combinatorial nomenclature previously introduced, we can remove two of the three binary naming states  reducing the system to two dynamic variables, Using model 2 as a basis, we can show that trans. The difference in Hill coefficient is then a consequence of difference in binding constants of the scaffold-associated versus scaffold-free MAPK enzymes. We would predict that the steric constraints of the fully assembled–but misaligned–scaffold would exhibit very weak enzymatic associations between the components of the pathway, leading to very large binding constants and therefore a very high Hill coefficient . On the other hand, if pure localization is the key determinant in ultrasensitivity, then this joint system will revert to a graded signal response.To test the hypothesis that it is not just membrane localization, but selective activation at the membrane, which leads to ultrasensitivity, we propose this experiment. Consider the situation of co-expression of the tethered Ste5 scaffold with the dominantly active Ste11. If our hypothesis is correct, this system will still demonstrate a strong ultrasensitive character, as there is no external signal through the receptor to permit selective activation. We cannot predict the exact value of the Hill coefficient, as there will be new competing forces acting on the system  to generate both an ultrasensitive (Fus3) and graded (Kss1) response. However, in their model, the scaffold itself is not addressed, which we can now rectify. This system is still dependent upon addition of pheromone, and so we assume the enzyme-scaffold binding reactions occur as described in model (2) previously. Once pheromone is added, the final complex formed involves active Ste11 and active Ste7. The issue is how this system is now less ultrasensitive than the wild type, as it involves all the key steps identified earlier as components of the ultrasensitive nature of the signal. The results in ncreased . Taken tTo test the plausibility of this hypothesis, we modify our model to ablate the fus3 binding site. We assume instead that fus3 is present in the cytoplasm and interacts via standard Michaelis-Menton kinetics with activated, scaffold-associated ste7. We then simulate the model under two conditions: first, without modification with a slow In this work, we have formulated two complimentary models of the yeast pheromone response pathway, in the absence (model (1)) and the presence (model (2)) of the Ste5 scaffold. The first model is a revisit of the original MAPK model first discussed by Ferrel, while the second is a simplified system describing the influence of the Ste5 scaffold on the MAPK pathway. We show how in both cases a careful examination of the binding constants, which dictate how strongly the enzymes associate with one another, lead to results consistent with recent experimented observations. In particular, we describe how manipulation of the protein-protein binding constants can lead to a multitude of signal response profiles, trending from strongly graded to sharply ultrasensitive.Specifically we highlight the following:Selective scaffold activation modulates signal output. Based on the results of our models, the dual requirements of Ste5 binding to the plasma membrane and phosphorylative activation of Ste11 to jointly induce MAPK signaling is crucial for allowing the system to exhibit both graded and ultrasensitive response profiles. The extent to which the selective activation step is able to shift the signal response from the scaffold ‘default’ of graded response to a switch-like ultrasensitive profile is critically dependent upon the protein-protein interaction strengths of the components in the system. In particular, stronger association of the scaffold with its activator decreases the ultrasensitivity of the system, as does stronger association of the selectively activated component with the scaffold.Prediction of global system changes based on local changes. The experimental papers under consideration in the formulation of these models both involved biochemical manipulations of the pheromone response pathway, leading to changes in the signal response. We have shown, by coupling the results of our simulations with the results of these manipulations, how mathematical models can be used to predict wide-ranging effects caused by these small perturbations of the original system. We have presented several hypotheses concerning how a particular change in the system–such as constitutive Ste11 activity or abrogation of Fus3-Ste5 interaction–leads to global changes in the system so that the observed signal response might be formed.A common criticism of mathematical analysis of biological networks is that, for standard analytic techniques to be applied, the system must be simplified to such a great extent that is must lose various important, complex details, rendering the results of the analysis suspect at best. However, we have shown how even a fairly simple model, model (2), is capable of suggesting general hypotheses about the nature of systemwide interactions based on a single perturbation event. This lends credence to the idea of intelligent reductionism, a feature of systems biology. Rather than reduce a complex system into all its possible individual components and study each in isolation, a complex system can instead be broken into reasonably independent modules, and each module studied alone and with regulatory interactions with other modules. This approach has been successfully applied to the study of receptor tyrosine kinases


The evolution of complex molecular traits such as disulphide bridges often requires multiple mutations. The intermediate steps in such evolutionary trajectories are likely to be selectively neutral or deleterious. Therefore, large populations and long times may be required to evolve such traits.We propose that errors in transcription and translation may allow selection for the intermediate mutations, if the final trait provides a large enough selective advantage. We test this hypothesis using a population based model of protein evolution.If an individual acquires one of two mutations needed for a novel trait, the second mutation can be introduced into the phenotype due to transcription and translation errors. If the novel trait is advantageous enough, the allele with only one mutation will spread through the population, even though the gene sequence does not yet code for the complettrait. Thus, errors allow protein sequences to "look-ahead" for a more direct path to a complex trait.This article was reviewed by Eugene Koonin, Subhajyoti De (nomimated by Madan Babu), and David Krakauer. E. coli cells appeared to acquire an excess of beneficial mutations. The idea that cells can somehow 'direct' evolution was thought provoking, and stimulated many investigations . Whiliews see ,7, the oiews see  have beephenotypic mutations) allow the selection of the intermediate mutations of a multiple-mutation requiring trait, and can thus speed up the evolution of complex traits.If mutations arise independently of their phenotypic consequences, then how can adaptations occur that require multiple amino acid mutations and for which the intermediate stages are either selectively neutral or disadvantageous? Large populations can climb multiple fitness peaks, even with disadvantageous intermediate alleles ,9. Altho-3 to 10-4 misreadings per codon k. Since the probability that allele 2 is generated k times follows a Poisson distribution with mean m, we find for the probability v that at least one of the mutations to allele 2 goes to fixationAssuming that  is u(s) . (In Appn(sβ) and nloss(sβ), setting m equal to either of these values. We assume that T is sufficiently large so that allele 1 has time to reach fixation within this interval (we assume T ≳ 2N). Then the probability u2 that allele 2 is generated and goes to fixation  isWe calculate this probability separately for u(sβ), is multiplied by the probability v that at least one copy of allele 2 is generated and fixed. The second half corresponds to the case of loss of allele 1 from the population, where the probability of loss of allele 1, (1 - u(sβ)), is multiplied by the probability of at least one mutation from allele 1 to allele 2 and subsequent fixation of allele 2. Taking into account allele 2 mutations during allele 1 loss is important especially for small s. Allele 1 is more likely to be lost than fixed for small s, but can occasionally drift for long times before being lost.The first half of the equation stems from the case when allele 1 eventually reaches fixation, where the probability that allele 1 becomes fixed, β → 0, i.e., in the absence of phenotypic mutations, we find with Eqs. (A2), (A27), and (A35)In the limit N ≫ 1, and neglect corrections of order 1 compared to N. Note that we cannot simplify (N + 1)/N to 1, because for small U, 1 - eNUu(s) -and (1 - eNU(T-N)u(s)-)/N are of the same order in N.) As we are interested in the effect of phenotypic mutations (β > 0) compared to the case without phenotypic mutations (β = 0), we define the increase in the probability of fixation from advantagous phenotypic mutations (the look-ahead effect) asu(s) -→ 1, in this limit, and thusWe can broaden the assumption of u(sβ)/u(0) = Nu(sβ).Apart from a correction for the case when allele 2 occurs while allele 1 is destined for extinction, Equation (5) is just the ratio of the probability of allele-1 fixation in the presence and absence of phenotypic mutations, sβ, Eq. (5) simplifies to (Appendix A.6)To first order in N is on the order of 1/(sβ).We can see from this equation that the look-ahead effect becomes important when Nsβ ≫ 1, only the first term contributes to the numerator in Eq. (5), and we obtain (Appendix A.7)For u2 and u2 by numerical simulation, for different values of s , (6) and (7). Figure s, the look-ahead effect can inflate the probability of fixation of allele 2 by several orders of magnitude. We also display the different analytic expressions for ξ in Figure T → ∞, works well for all values of s. The approximation (6), derived for small sβ, captures correctly the magnitude of s at which the look-ahead effect starts to operate, i.e., s ≳ 1/(Nβ). Similarly, approximation (7), valid for Nsβ ≫ 1, approximates ξ well for larger s.We confirmed our analytic results for the fixation probabilities ξ for different population sizes. As expected from the condition s ≳ 1/(Nβ), the look-ahead effect will work with smaller selection coefficients s in larger populations. For large s, ξ saturates at approximately N.Figure β  would be neutral. We refer to selection for the intermediate allele as the sβ, Eq.(6), shows most clearly the relationship between the parameters. The look-ahead effect is proportional to N, s, and β, and sets in when N is on the order of 1/(sβ). For large Nsβ, the look-ahead effect saturates. The asymptotic value of ξ is approximately N for NU ≪ 1.The approximation for small s, and saturates at a larger asymptotic value ξ. Of course, even in the absence of the look-ahead effect, larger populations can more easily traverse multiple local fitness peaks  for alleles , giving the new number of alleles in a possibly larger population. Finally, the next population of N individuals is chosen by recursively sampling from the binomial distribution, representing random genetic drift. Allele 0 is first sampled with the mean = , and the  = N. Allele 1 is then sampled from the combined allele 1 and 2 individuals. The number of simulations where allele 2 becomes fixed is divided by the total number of simulations, giving an estimate of the fixation probability. The number of simulations for each parameter set was between 108 and 109.The population in each simulation is represented by three numbers, corresponding to the abundance of each of the three alleles. As described, the initial abundances are DJW, DV, COW, and EBB developed the original idea, DJW and COW performed the simulations and analysis, all authors contributed to the writing.Eugene V. Koonin, NCBI, NLM, NIH, Bethesda, MD 20894, United StatesThe idea of this paper is as brilliant as it is pretty obvious...in retrospect. A novel solution is offered to the old enigma of the evolution of complex features in proteins that require two or more mutations (emergence of a disulphide bond is a straightforward example). Whitehead et al. propose that selection for such traits could be facilitated by phenotypic mutations . Due to phenotypic mutations, rare variants of proteins will emerge that are "pre-adapted" to accommodate the second, beneficial mutation, yielding the complex, adaptive trait, even if transiently. Simply put, for the case of a disulfide bond, one cysteine appears as a result of a phenotypic mutation and the other one due to a genotypic mutation. The result will be that, for a while, the cell will have in its possession the protein molecule with a disulfide bond. Thus, "pre-adaptation" owing to phenotypic mutation would promote fixation of the second mutation which will be beneficial even without the first one – if the selective advantage of the complex trait is high enough . The actual fixation of the complex trait, then, requires only one (the first) mutation and is thus greatly facilitated. Mathematical modeling described in the paper shows that, if the selective advantage of the complex trait, i.e., the selection coefficient for the second mutation, is high enough, this look-ahead effect becomes realistic under the experimentally determined mistranslation rates. Obviously, the realization of the look-ahead effect will depend on a variety of factors including the overall translation fidelity, the local context of the codon involved, the stability of the protein etc. This allows a number of rather straightforward experimental tests of the model.From my perspective, this is a genuinely important work that introduces a new and potentially major mechanism of evolution and, in a sense, overturns the old adage of evolution having no foresight. It seems like, even if non-specifically and unwittingly, some foresight might be involved. At a more general conceptual level, this work is important in that it puts together, within a single conceptual framework, the evolutionary effects of genotypic and phenotypic mutations. There is much more to investigate here!I would like to mention a rather general biological implication. It seems obvious enough that, under conditions of stress , when translation fidelity drops, the look-ahead effect will be enhanced. Thus, this could be a general and crucial mechanism of adaptation during evolution.Eugene KooninAuthor response: We would like to thank Eugene Koonin for his enthusiastic and positive review.Subhajyoti De, MRC Laboratory of Molecular Biology Hills Road, Cambridge CB2 2QH, United KingdomI have read the revised manuscript, and have found that all points raised by the referees were fully addressed. The work is rigorous and very interesting, and I believe, will make a significant contribution in the field. I'll be happy to consider it for publication.Subhajyoti DeAuthor response: We would like to thank Subhajyoti De for feedback that improved the original manuscript, and the subsequent positive review.David Krakauer, Santa Fe Institute, United StatesIn this paper the authors demonstrate how phenotypic variation arising through errors in development (e.g. transcription and translation), can, when building on (amplifying) genetic variation, accelerate the fixation rate of neutral alleles. By assuming that neutral alleles are genetically closer to an optimum genotype than a mutation-free wild-type, this can also reduce the time required to reach the optimum. The result is illustrated through stochastic simulation and some limiting-case analytical approximations.This is an interesting paper that is technically rigorous, and correct in many of the conclusions that it reaches. The paper is now much improved as it now includes specific reference to the almost identical, Baldwin effect. As the authors correctly state, many of papers on the Baldwin effect emphasize learning, but a significant fraction explore the role of random ontogenetic variation on evolutionary dynamics, and a few, explicitly consider the adaptive value of errors in transcription and translation on the exploration of fitness landscapes. It is not yet clear how important the differences are between treating Baldwin effects in terms of individual ontogenetic programs versus population level dynamics. In both cases, the key insight is that random variation is capable of generating a more effective gradient for population dynamics.I think it worthwhile therefore to give a brief review of this mechanism and a little of its literature.A Synoptic Outline Of the Baldwin-Morgan-Osborn Effect1. The essential insight of Baldwin and several other 19th century biologists (listed above) was to understand that phenotypic plasticity can have a direct effect on genetic evolution. In some cases, this can give rise to the appearance of Lamarckian inheritance, as selection on plastic phenotypes derived from a single genotype, can lead to the fixation of polymorphic sequences generating these phenotypes without plasticity.2. The modern investigation of this effect is associated with the work of Hinton and Nowlan (1987) who showed that ontogenetic variability or plasticity, could lead to effective genetic optimization in neutral fitness landscapes.3. This has been followed by numerous papers exploring complex landscapes, diverse models of plasticity, including learning, homeostasis, diffusion, and combinatorial sampling. See Turney (1996) for a review with an emphasis on computational approaches.4. Ancel and Fontana (2000)  demonstrated for RNA secondary structure, the crucial requirement that phenotypic plasticity and genetic polymorphism should exhibit a particular correlational structure for the Baldwin effect to be effective.5. The most recent, and somewhat exhaustive analysis of the Baldwin effect has been conducted by Borenstein et al (2006) in fluctuating landscapes, exploring both directed and random phenotypic variation.6. Krakauer and Sasaki (2002) demonstrated a "negative Baldwin effect" whereby developmental errors could amplify mildly deleterious mutations in finite populations, thereby leading to their effective purging.Certainly the paper by Krakauer and Sasaki does not consider learning explicitly, but something much closer to the so called "look ahead effect" described by Whitehead et al, as it treats the ensemble of variant proteins generated by a single underlying sequence as a result of errors in transcription or translation. In both the Baldwin effect and the "look ahead" effect, genetically identical organisms generate phenotypically diverse populations. I think it an interesting subject for future work to establish the precise nature of any differences manifesting at the level of population dynamics, rather than at the incidental level, of mechanism.Author response: We appreciate this correction of a large hole in our background literature. We have cited relevant literature about the Baldwin effect, and discussed the main differences between the look-ahead effect and the Baldwin effect. While on the surface the look-ahead effect is very similar to the Baldwin effect, crucially the Baldwin effect is about individual learning, whereas the look-ahead effect is about errors that always produce different proteins from a single gene, at a given rate. Thus, in our model there is little difference between individuals with the same genotype, as no learning is involved, as opposed to the Baldwin effect, where, due to learning, two organisms with identical genotypes can have very different phenotypes. Therefore, we believe that it is important to distinguish clearly between the cases with and without learning, and to use different terminology to emphasize this distinction.I was somewhat confused by the remark that double mutations are neglected because they are very rare.Firstly, double mutations should be allowed within the binomial model presented by the authors. Secondly, the statement is empirically false for many haploid genomes. Bonhoeffer and Nowak (1997) showed that in large populations double mutants are likely to exist at fairly high abundance.Author response: We agree that for RNA-based viral genomes, which often have genomic mutation rates 1000 times greater than DNA-based organisms, double mutations occur frequently. Our model focused on DNA-based organisms, where double mutations are rare. If we wanted to apply our model to RNA viruses, we would have to include double mutations. However, the results from such a modification are obvious: If double mutations are frequent, the organism will happen upon the beneficial double mutation quickly and not require the look-ahead effect at all.The treatment of deleterious mutations remains a little confusing. Presumably developmental noise can both amplify existing deleterious effects  and contribute novel pathologies, orthogonal to those of the underlying transcript (e.g. gain of function mutations). This should be made an explicit, distributional property of the model rather than assuming a fixed background cost.Author response: The explanation of how we treat deleterious mutations was extremely brief in our original draft, and we have expanded and clarified the respective paragraph. We believe that a more explicit, complex treatment of deleterious effects would detract from the main message the model in this work was meant to convey. We have added to the discussion how phenotypic mutations can amplify deleterious genotypic mutations. A more complete treatment of deleterious phenotypic mutations will be a topic of future work.Here, we present the details of our analytic derivations.u(s) of a single allele with selection coefficient s is given byAccording to , the pros ≳ 1/N, this expression simplifies toFor Ns ≫ 1, this expression simplifies towhereas for s drifting to fixation or extinction, and ask how many mutations this allele generates until it is either fixed or lost. We will treat these two cases separately. Let nfix(s) be the expected number of mutations generated while the allele drifts to fixation, and let nloss(s) be the expected number of mutations generated while the allele drifts to extinction. We calculate these two quantities using diffusion theory, by integrating the sojourn times of the allele over all frequencies.We first consider a single allele with selective advantage s and starting at frequency p = 1/N, [This expansion follows directly from the definitions of Chi = nfix(s) + nT(s), where nT(s) is the total number of mutations generated once the first mutation has reached fixation. We haveWe calculate tfix(s) is the time to fixation of a mutation with selective advantage s. This time is given by the integral over all sojourn times,where withI2(a) = 2I(a), and thus we haveA partial fraction decomposition of the integrand reveals that Combining this result with Eqs. (A13) and (A30), we findn(s) = nfix(s) for T = tfix(s).Note that s ≲ 1/N, we findFor Ns ≫ 1, using Eqs. (A15) and (A19), we obtain the asymptotic expansionFor sβFrom Eq. (4), using Eqs. (A27), (A35), and (A2), we obtain to first order in NU(T - N)u(s) ≪ 1, we obtainIf further T → ∞,and for Nsβ ≫ 1, only the first term contributes to Eq. (2), and we obtain from Eqs. (A36) and (A3)For Likewise, in this limit we can simplify Eq. (3) togivingT → ∞, this expression simplifies toFurthermore, for NU ≪ 1, then ξ → N in the limit s → ∞.If


Recent evidence suggests that mutation rates are fitness-dependent, broadening our view of the impacts of mutation on the genetic health of populations.  Many a research paper, textbook chapter, and grant proposal has begun with the phrase “Mutation is the ultimate source of genetic variation.” Implicit in this phrase is the assumption that genetic variation is required for evolution. Without mutation, evolution would not be possible, and life itself could never have arisen in the first place. However, there is overwhelming evidence that the great majority of mutations with detectable effects are harmful . DeleterDeleterious mutations are known or thought to influence a wide variety of biological phenomena  for a comprehensive account), the most notable of which is sexual reproduction. To make a very long story short, there are several reasons that asexually reproducing taxa are expected to have a short-term evolutionary advantage over sexually reproducing taxa. Specifically, the fact that a gene is passed to all the offspring of an asexual individual and to only half the offspring of a sexually reproducing individual provides a 2-fold fitness advantage to a new mutation that confers asexual reproduction when it arises in a population of sexual organisms. This 2-fold difference in fitness is known as the “2-fold cost of meiosis” . Howeversine qua non of sex is genetic recombination, and deleterious mutations are more readily removed from a population in the presence of recombination than in its absence. A large body of theory predicts that, all else equal, the greater the mutation rate, the greater the probability that sexual reproduction will be favored. However, recent research by Aneil Agrawal [Understanding why and how sexual taxa have managed to consistently overcome the 2-fold cost has probably absorbed more intellectual energy than any other single problem in evolutionary biology. Many arguments have been advanced to explain the prevalence of sexual reproduction ,6, but t Agrawal ,11 callsEscherichia coli have higher mutation rates when starved . The bamstances .Importantly, the adaptive mutation scenario is only plausible in taxa that reproduce primarily asexually, because recombination breaks up the association between the alleles that influence mutation rate and alleles that influence fitness ,17. AnotPoor physiological condition may occur for two fundamental reasons, environment and/or genes. A simple example of the former is genetically identical plants raised under variable conditions of moisture: plants that are watered will be in better condition than those that are not. An example of the latter is harder to contrive, but it has been shown that under identical environmental conditions, there is genetic variation for fitness. Variation in fitness could be due either to variation in the number of deleterious alleles an individual carries in its genome or variation in the effects on fitness, or both. If individuals carrying more deleterious alleles tend to be in poor condition and if individuals in poor condition tend to have higher mutation rates, the existence of a positive-feedback process is suggested, which leads to an upwardly spiraling mutation rate and downwardly spiraling fitness.U), which is assumed to be constant, and is independent of the strength of selection [−U and the genetic load is 1 − e−U; this result holds for both sexual and asexual populations, in the absence of nonadditive effects [A useful way of quantifying population-genetic phenomena is in terms of the “genetic load,” the reduction in fitness of a population of interest relative to a population composed solely of the most-fit genotype. Of particular interest is the genetic load at mutation-selection balance (MSB), the point at which the input of genetic variation from mutation is exactly balanced by the removal of deleterious mutant alleles by natural selection. A classic, perhaps surprising result is that the genetic load at equilibrium in an infinitely large population is determined solely by the genomic deleterious mutation rate , with individuals in the best-possible condition mutating at rate UMIN and those in the worst-possible condition (other than dead) at UMAX. The functional relationship between fitness and U is assumed monotonic but not necessarily linear, and the fitness effects of mutations are assumed multiplicative across loci.A functional relationship between fitness and mutation rate complicates the situation. The first theoretical treatment of fitness-dependent mutation rates was given by Agrawal  for the asex ≈eUMIN−, even though most genotypes in the population may have many more than zero mutations. Second, the equilibrium fitness of a sexual population depends critically on the nature of the functional relationship between mutation rate and fitness (sex will approach that expected at the maximum mutation rate (UMAX). Conversely, if the mutation rate is a concave upward (accelerating) function of genetic load, Ŵsex will be much closer to that expected at the minimum possible mutation rate (UMIN).Two important proximate results emerge from Agrawal's analysis. First, for an asexual population, mean fitness at equilibrium is a function of the mutation rate of the genotype with the fewest deleterious alleles . If the most-fit genotype has zero deleterious mutations, Ŵ fitness . If the This analysis points to a key difference in the equilibrium genetic load in sexual and asexual populations under constant and fitness-dependent mutation rates. If the mutation rate is constant, then sexual and asexual populations are expected to have the same genetic load at equilibrium. In contrast, if the mutation rate is fitness-dependent, the expected fitness in an asexual population at equilibrium will be that of the most-fit genotype, while in a sexual population, the equilibrium fitness will be less than that of the most-fit genotype, perhaps much less. Thus, surprisingly, the “cost of sex” may in fact be much greater than 2-fold.Chlamydomonas [Drosophila melanogaster that had been allowed to accumulate mutations for 160 generations increased 2.5-fold over the ancestral stock. That result is consistent with an increase in the mutation rate with genetic load (as claimed by the authors in the title of the article), but it is also consistent with an increase in the average deleterious effect of a new mutation with increasing genetic load, i.e., “synergistic” epistasis [Although condition-dependent, inducible increases in mutation rate are well-documented in microbes, including yeast  and the ydomonas , there iydomonas  showed tpistasis .PLoS Biology, Agrawal and Wang report a study designed to determine if the mutation rate is condition-dependent in D. melanogaster without the ambiguity associated with classical mutation accumulation studies [D. melanogaster, but DNA damage carried in sperm can be repaired post-fertilization in D. melanogaster embryos via maternal repair enzymes. Manipulating female condition could alter maternal repair processes. There are multiple maternal repair pathways, some of which are error prone and some of which are not. The error-prone pathways are believed to be less metabolically costly. Agrawal and Wang hypothesized that females in poor condition would preferentially use the less-costly error-prone mechanisms, whereas females in good condition would preferentially use the more-costly high-fidelity mechanisms. Condition was manipulated by diet; genetically identical females were fed high- or low-quality diets and fertilized with sperm that had been subject to mutagenesis. The relative frequency of recessive lethal alleles was 30% greater on paternally derived X chromosomes that had passed through low-condition females rather than high-condition females.In this issue of  studies . Their eDo these results prove that the mutation rate is condition-dependent? Not quite. It remains to be shown that the spontaneous mutation rate is condition-dependent. However, if the efficacy of DNA repair is in general condition-dependent, a lack of condition-dependence of the spontaneous mutation rate would require there be a compensatory reduction in the probability that DNA damage occurs. Nevertheless, this study provides the strongest evidence yet for condition-dependent mutation rate in a multicellular eukaryote.reduce the mutation load).Condition-  dependent mutation leads to some interesting possibilities for the long-term genetic health of populations, including our own species. For example, it has been argued that modern technology  has led to the relaxation of selection against mildly deleterious mutations in the developed world, leading to a build-up of genetic load ,25. If tA second interesting possibility is that condition-dependent mutation could, in effect, render temporary increases in mutagenesis due to environmental causes permanent. Many anthropogenic factors are known to be mutagenic, not only to humans but to many other organisms. If the mutation rate is condition-dependent, a short-term increase in the input of mutation due to (say) a mutagenic pollutant could lead to a long-term increase in the mutation rate, and thus in the genetic load. Similarly, genetic drift allows the fixation of slightly deleterious alleles in small populations. With condition-dependent mutation rate, a temporary bottleneck in population size that results in the increase in frequency of deleterious alleles could lead to an effectively permanent increase in the mutation rate.


Mutational fitness effects can be measured with relatively high accuracy in viruses due to their small genome size, which facilitates full-length sequencing and genetic manipulation. Previous work has shown that animal and plant RNA viruses are very sensitive to mutation. Here, we characterize mutational fitness effects in single-stranded (ss) DNA and ssRNA bacterial viruses. First, we performed a mutation-accumulation experiment in which we subjected three ssDNA  and three ssRNA phages  to plaque-to-plaque transfers and chemical mutagenesis. Genome sequencing and growth assays indicated that the average fitness effect of the accumulated mutations was similar in the two groups. Second, we used site-directed mutagenesis to obtain 45 clones of ΦX174 and 42 clones of Qβ carrying random single-nucleotide substitutions and assayed them for fitness. In ΦX174, 20% of such mutations were lethal, whereas viable ones reduced fitness by 13% on average. In Qβ, these figures were 29% and 10%, respectively. It seems therefore that high mutational sensitivity is a general property of viruses with small genomes, including those infecting animals, plants, and bacteria. Mutational fitness effects are important for understanding processes of fitness decline, but also of neutral evolution and adaptation. As such, these findings can contribute to explain the evolution of ssDNA and ssRNA viruses. The fitness effects of mutations are the raw material for natural selection. It has been shown that point mutations typically have strongly deleterious effects in plant and animal RNA viruses, whereas cellular organisms are comparatively more robust. Here, we characterize the fitness effects of random mutations in DNA viruses and compare them with those found in RNA viruses, using six phage species of similar genome sizes. To achieve this goal, we introduced mutations by chemical and site-directed mutagenesis, identified the genetic changes by sequencing, and quantified their fitness effects using growth-rate assays. In all cases, mutations had a strong average impact on fitness. We conclude that mutational sensitivity is a general property of viruses with small genomes and discuss the evolutionary implications of these findings. Mutational fitness effects are important for understanding the genetic variability of populations, the relative roles of natural selection and drift, the origin of sex and recombination, or the ability to produce evolutionary innovations, among other processes Escherichia coli more than 90% of gene knock-outs are viable Some progress has been made in characterizing mutational fitness effects. For instance, in Viruses are a unique experimental system for characterizing mutational fitness effects because it is relatively easy to engineer sets of random single-nucleotide substitutions. This has been done previously for single-stranded (ss) RNA viruses, but not for ssDNA viruses. These two types of viruses have similar genome sizes and therefore might be expected to be equally sensitive to mutation. On the other hand, given the obvious ecological E. coli strain under identical environmental conditions Here, we compare mutational fitness effects in ssDNA and ssRNA viruses using six phages which can infect the same Preliminarily, we adapted each phage to our laboratory conditions by performing serial passages at high population sizes, until all lineages reached stable fitness values. For each phage, three independent mutation-accumulation lineages were initiated by picking single plaques at random from the adapted populations. Phage supernatants from plaques were subjected to chemical mutagenesis using nitrous acid and plated to isolate new plaques, a protocol that was continued until plaque sizes became drastically reduced. Under these conditions, selection is minimized and therefore, except for highly deleterious or lethal mutations, random genetic drift and mutational pressure are the only factors driving the fixation of mutations P = 0.769) and there were no significant differences between the DNA and RNA groups (P = 0.605). After sequencing the six ancestral and the 18 evolved genomes (GenBank accession numbers GQ153912-GQ153935) we found that DNA and RNA phages had accumulated 157±3 and 146±1 nucleotide substitutions in total, respectively (P = 0.802). We calculated the per-mutation effect by dividing the fitness loss of each lineage by the number of mutations accumulated and again, this did not reveal significant differences between DNA and RNA phages (P = 0.870). We therefore conclude that selection strength is similar in the two groups. F1 appeared to show the highest level of robustness, but differences between species were non-significant (P = 0.088) and a Tukey's post-hoc test indicated that the six phages formed a single coherent group.The change in fitness was homogeneous among species (P = 0.010). This might be due to their lower spontaneous mutation rates or to lower susceptibility to the mutagen. In both groups, the proportion of transitions was high, and this excess was significantly more marked in DNA phages . Since nitrous acid induces mainly transitions, this suggests that, in DNA phages, most substitutions were caused by the mutagen, whereas in RNA phages there was a greater contribution of spontaneous mutation. This could explain why more passages were required in the former group, although differences in susceptibility to the mutagen or in the proportion of transitions among spontaneous mutations cannot be discarded. Importantly, the higher transition/transversion ratio of DNA phages did not result in a significantly higher fraction of synonymous substitutions  and therefore is unlikely to have biased the above selection strength estimates.However, the number of passages required to reach similar fitness losses or mutation numbers was higher on average for DNA phages . All lethal mutations found in ΦX174 produced amino-acid changes, whereas in the case of Qβ, one was synonymous (U2379A) and one intergenic (G1329A).We constructed 45 clones of ΦX174 and 42 clones of Qβ carrying single-nucleotide substitutions, choosing the target site and the substitution at random. First, we amplified the viral genomes by PCR using mutagenic primers, transfected ectively , i.e. a V(s) = 0.162. For viable mutations only, V(vs) = 0.047. In Qβ, V(s) = 0.181, V(vs) = 0.018 . The above correction using controls implies that our inferences of the mean and variance should be free of experimental error or bias resulting from the presence of additional mutations or changes in the assay conditions, although it is still possible that the fitness values of some individual mutants were altered by the presence of additional mutations.We measured the growth rate of viruses carrying viable substitutions to obtain the distribution of mutational fitness effects. Growth rates were also determined for the above control samples and corrected means and variances were obtained by subtracting the mean and variance of the control group from the grand mean and variance of the mutants. In ΦX174, this yielded a mean selection strength of  = 0.018 . There ws-values, we performed non-linear regression to characterize the shape of the distribution. In ΦX174, an exponential model with an expected mean selection strength against deleterious mutations of R2 = 0.989) and was as accurate as a Gamma (R2 = 0.989) or a Beta (R2 = 0.985) model . Other statistical models were not considered here. In Qβ, the data were better described by the Gamma distribution (R2 = 0.975) with an expected R2 = 0.973), whereas the exponential model was slightly worse . Finally, we must emphasize that mutants with positive values were not used for these inferences, which implies that they were assumed to be neutral or beneficial. Neglecting beneficial mutations, the fraction of deleterious to total mutations should thus be dp =  (lp)/ds and the fraction of neutral to total mutations simply np = 1−lp−dp , We analyzed ΦX174 using the same methodology and under the same environment as Qβ to make the comparison between ssDNA and ssRNA viruses. The two phages did not significantly differ in the fraction of lethal mutations or the average strength of selection, suggesting that ssDNA and ssRNA viruses are similarly sensitive to mutation. The strong parallel shown by viruses as different as ΦX174 and Qβ probably stems from the fact that both have small genomes with few and short non-coding regions, several multifunctional proteins, and little genetic redundancy The similar robustness shown by ssDNA and ssRNA viruses was also supported by the mutation-accumulation experiment, in which we examined three phage species of each group and found no significant differences in mean selection strength. Differences between species within the two groups were weak or absent, although F1 appeared to be the most robust phage. Related to this, it is worth noting that F1 has fewer overlapping genes. This might be related to the fact that inoviruses have filamentous capsids, which are structurally less constrained than icosahedral ones and can hence accommodate larger genomes It is possible to compare directly the mean selection strengths estimated by mutation-accumulation and site-directed mutagenesis, since the same viruses and the same environment were used. Because lethal mutations cannot be sampled during plaque-to-plaque passages we expected the former to be lower. However, even after excluding lethals, the The fitness effects of random mutations are relevant to evolution in many ways. For instance, they determine the fraction of nucleotide sites that evolves neutrally and thus the rate at which populations diverge through random genetic drift. Further, neutral and deleterious mutations can also determine the rate of adaptive evolution indirectly. The observation that mutations tend to be highly deleterious indicates that there are no efficient buffering mechanisms, which might result in large phenotypic variation and strong selection for beneficial mutations too The fitness effects of random mutations are not only relevant to neutral evolution and adaptation, but obviously, also to processes of fitness decline such as Muller's ratchet A conceptual dichotomy between DNA and RNA viruses has been established based on the observation that the latter evolve faster E. coli C strain IJ1862 Bacteriophages ΦX174, G4, F1, Qβ, SP, and MS2 and the 5 particle forming units (pfu) were inoculated into 0.5 mL LB medium containing IJ1862 cells at their exponential growth phase. The appropriate cell density varied depending on the virus lytic activity and growth rate . Infected cultures were incubated in agitation (650 rpm) at 37°C in a Thermomixer 24-tube shaker (Eppendorf) and harvested during the late exponential growth phase of the virus (∼109 pfu/mL). Cells were removed by centrifugation and supernatants were aliquoted, stored at −70°C, and titrated using LB medium solidified with soft agar (0.7%). Initial and final titers were used to calculate growth rates and to adjust sampling times for the next passage accordingly. We performed 60–80 passages for each phage. In all cases, no significant changes in growth rate were observed during the last 30 passages, indicating that phages had reached a quasi-maximal fitness under these experimental conditions.Each phage was serially passaged at high population sizes in IJ1862 cells. For each transfer, ∼10Three independent mutation-accumulation lines were seeded for each phage by picking random plaques from the high-fitness populations. Each plaque was resuspended in 50 µL LB and stored at −70°C. Lines were propagated plaque-to-plaque as previously described E. coli IJ1862 cells and from an overnight culture of E. coli K12 previously transformed with this cDNA, respectively. For PCR-based mutagenesis, full-length PCR amplicons were obtained from 500 pg of template using Phusion high-fidelity DNA polymerase  and a pair of adjacent, divergent, 5′ phosphorylated primers, one of which carried the desired nucleotide substitution. PCR products were circularized using the Quick T4 ligase (New England Biolabs) and E. coli IJ1862 competent cells were transfected by the heat-shock method  in the presence of CaCl2 100 mM. The transfected cells were immediately plated onto LB plates using soft agar and individual plaques were picked after 5-9 hours of incubation at 37°C, resuspended in LB, and stored at −70°C. To verify that the desired mutation had been introduced and that no additional changes were present in the region flanking the target site, (RT)-PCR was performed directly from the resuspended plaque. Moloney murine leukemia virus reverse transcriptase (Fermentas) was used for cDNA synthesis and Phusion DNA polymerase for PCR. The products were column-purified and sequenced using sequence-specific primers.The High Pure Plasmid Isolation kit (Roche) was used to purify ΦX174 DNA and the pBRT7β plasmid from partially lysed In cases where transfection yielded no plaques or phages recovered from plaques had not incorporated the mutation, the entire protocol was repeated and, after three consecutive failures, the mutation was classified as a candidate lethal. In all of these cases, the number of plaques obtained from transfection assays was abnormally low, suggesting that the mutation was lethal for the virus and that the few observed plaques came from the template wild-type DNA. To confirm lethality, we first sequenced the region of the PCR-based mutagenesis product flanking the target site to check that the mutation was present and that no additional changes had appeared in this region. Then, we designed new primers that did not carry the nucleotide substitution but were otherwise identical to those used for the mutagenesis reaction, and performed the PCR, circularization and transfection steps exactly as above. Lethality was assessed based on the comparison between the numbers of plaques recovered from mutagenesis versus control reactions .Similar control assays were carried out to estimate the contribution of non-desired mutations and other sources of error to the inferred distribution of mutational fitness effects. For each phage, we transfected three of the above control PCR-based mutagenesis products and picked 11–12 random plaques from each, yielding 36 and 33 plaques in total for ΦX174 and Qβ, respectively. The relative fitness of each clone was determined as above and we obtained the mean and variance for each phage. These values were subtracted from the means and variances obtained for the real mutants. This allowed us to account for experimental error and, in particular, to control for any potential mutations present in the DNA templates, arising during PCR amplification, or during plaque growth.4 pfu were inoculated into 0.5 mL of LB medium containing IJ1862 cells at their exponential growth phase. Infected cultures were incubated in agitation (650 rpm) at 37°C in a Thermomixer 24-tube shaker and harvested when the wild-type reached an estimated titer of 108 pfu/mL. Cells were removed by centrifugation and the supernatants were aliquoted, stored at −70°C and titrated. The growth rate (r) was calculated as the increase in log-titer per hour. Relative fitness was obtained as W = ir/0r, where i and 0 refer to the mutant and wild-type, respectively , and the fitness effect (selection coefficient) was calculated as s = W−1. In all cases, mutant and wild-type samples were assayed in the same experimental block, and experimental blocks were done in triplicate.To measure growth rates, ∼10Sequences were obtained using the Sanger method (Applied Biosystems). In general, we sequenced column-purified plaque-(RT)-PCRs directly. In cases where this yielded low-quality readings, we cloned the PCR fragments using the Zero Blunt TOPO PCR Cloning kit (Invitrogen) and sequenced the inserts with vector-based and internal primers, discarding mutations that were not present in at least 3/5 clones.v, the model can be written as ijkν = μ+iG+ijS+ijkl, where μ is the grand mean and l is the line (error term).For the mutation-accumulation experiment, we used a univariate linear model with two factors: Genetic material  and Phage species , whereas the line was the experimental replicate. The variables analyzed were the number of passages, the number of mutations fixed, and the expected fitness effect per mutation. For each variable ijkls = μ+iS+ijA+ijkP+ijklε, where μ is the grand mean, ε is the error term.To test for differences in average fitness between the ΦX174 and Qβ mutant collections, we used a univariate linear model with three factors: Phage species , Assay type , and Plaque . Hence, each experimental determination can be expressed as V(ds) using the Levenberg-Marquardt algorithm implemented in SPSS v12.Non-linear regressions were performed to estimate Figure S1E. coli cells with these PCR producuts are shown below the brackets. (A) For ΦX174, all mutations except A402T were confirmed as lethal. (B) For Qβ, all mutations except G780U, G1110A, and G2587U were confirmed.Confirmation of lethal mutations. Site-directed mutagenesis reactions and control reactions  were loaded in 0.8% agarose gels and stained with ethidium bromide. For each candidate mutation, the left and right lanes contain the mutagenesis and control reactions, respectively. Bands of the expected size (5.4 kb for ΦX174 and ca. 7.0 kb for Qβ insert plus the vector) were obtained in all cases . The number of plaques obtained after transfection of (0.43 MB PDF)Click here for additional data file.Table S1Genetic changes found in ΦX174, G4, F1, Qβ, SP and MS2 mutation accumulation lines (three lines per phage). The final passage number, the relative fitness and its standard error (SEM), the nucleotide substitutions found (genomic position and substitution), the genes where they map, and the associated amino acid change (amino acid number within gene) are shown. Notice that some mutations fall at overlapping genes (the respective amino acid substitutions are indicated in these cases). Genomic positions in ΦX174 are numbered following the convention that nucleotide 1 corresponds to the Pst I cleavage site. Sequences covered nearly the whole genome (>98%) in all cases except for line SP.1 in which we failed to sequence genomic regions 1–167, 653–823, 2277–2633, and 2985–3186, and line MS2.3 in which we failed to sequence region 3048–3569.(0.03 MB PDF)Click here for additional data file.Table S2ΦX174 point mutants obtained by site-directed mutagenesis. For each mutant, the genomic position of the nucleotide substitution (relative to the Pst I cleavage site), the mutated gene, the amino acid change, and the relative fitness effect ± SEM are shown. Assays were done in triplicate and fitness values were corrected to account for the presence of additional mutations and other sources of experimental error see . Notice (0.02 MB PDF)Click here for additional data file.Table S3Qβ point mutants obtained by site-directed mutagenesis. For each mutant, the genomic position of the nucleotide substitution, the mutated gene, the amino acid change, and the relative fitness effect ± SEM are shown. Assays were done in triplicate and fitness values were corrected to account for the presence of additional mutations and other sources of experimental error see . Notice (0.02 MB PDF)Click here for additional data file.