数理解析研究所講究録 1420 巻 2005 年 1-17

1Interation states by iteratrng maps
in molecular biology
Tsuyoshi KatoIntroduction
Biological systems a.re consisted by hierarchy of many kinds of systems
with very different time scalings each other. Soinetimes this causes animportant phenomena, namely random interaction in micro scale can form
several patterns in macro. Here one wants to try to express them by using
the dynamical systems of iterating maps.
In part I, we will describe which kinds of properties in micro scopic interaction systems nlight occur when one uses iteration of rnaps. Molecularinteraction systems are equipped with various types of operations, for example switching of genes, various kinds of cycles including biological clocks,
and mutations. We will clescribe how such phenolnena rnight occur in thesystems of iterations of znaps.
This part mainly consistes of explanatiozr of biological phenom ena byinterpreting them into several properties of iteration of maps, which also
includes suggestion of direction of mathematical formulation rigorously.
In part Ij we will formulate micro interaction by use of composition of
nlalps, which is used in [K2]. There we have a realization of some parts ofthe description in part I in a rigorous way. In particular we have studied
how random micro interaction defined by this way can form several patterns
in more macroscopic scales. In part II, we state shortly some results from[K2].
I A description of interaction phenomena

1

Feedback system-ntechanisms as molecules, like ensymes,
their dynarnics seem to hold botlr aspects, deterrninistic and non-deterministic
behaviours. The forlller is because many proteins are polymers and whose
dynamics apply classical lnechanics. Quantum effect by electron enters mto

When proteins play roles in$\mathrm{b}\mathrm{i}\mathrm{o}$2the latter. In particular in chemical reaction, molecular oscillation palys avery important role. One im portant point is that these have very different time scalings mutually. Here one will try to represent bio-interaction
systems by taking into account of these features.
Let : $[0, 1]arrow[0,1]$ be a map. In general structure of the range will bevery com plicated. A point $x\in[0,1]$ will represent a state of a molecule.
An oscillation of is the orbit:
$f$

$x$

$\{f^{??}(x)\}_{\tau\prime=0_{\backslash }1_{\backslash }}\ldots=x,$$f(x),$ $farrow(x)?,$$\ldots$

Dynamics of a molecule is a long-tim le behaviour of the oscillations. We
are mainly interested in behaviour of states around period orbits:$P(?\iota)=P(?\mathrm{z})(f)$ $=\{x\in[0,1] :f’{}^{\mathrm{t}}(x)=x.\}$.

When a molecule is interacting with other molecules or is of phase transition, then its oscillation, or hybridization shows chaostic behaviour inquantum chem istry. An element $x\in[0,1]$ near the set $P(n)$ will behave
either stable or unstable manners. Unstable points will cause chaostic phenomena. $x\in P(n)$ is regarded as a high state, if 7? is large. When is asmall periodic point, then it is in a Iow state.
In order to consider dynamics of molecules, here one uses sim ple differential equations including time parameter. This parameter corresponds tothe large scaling compared with oscillations of molecules.
Now take a smooth function :which could show gradient ofenergy functional at a state $x\in[0,1]$ . With respect to the time parameter,
feedback equatzon for a single state is written as:$x$$l\mathrm{t}$$[0, 1]arrow \mathrm{R}$$\dot{x}\cdot=-h(x)$.

This represents a large scale dynamics, and we need to take into accountof small scale oscillations.
Molecules can change the states into very different quantum states. Thisis the tunnel effect. One way to express this phenomena is to use theenergy potential surfaces by introciucing extra parameter (e.g. complex
coordinate).
One may construct usinq th le energy potential surfaces. On the plane$[0, 1]$, one plots a point
for each periodic point $x\in[0,1]$ withperiod with respect to . If one requires some Gaussian-like shapes for
mountains on the plane, then one will need to change scale of coordinate,
say $xarrow x’$ which may be used as $h(x)=x’$ .
$h$$\mathrm{x}\mathrm{R}$$y$$(\alpha\cdot, y)$

$f$

$x$

3
2

Time scaling

In general tim scalings are completely different between reaction velocity
of ensymes and oscillation of the molecules. When genes are interacting,one cannot choose one state uniquely for each gene. Thus one is inebitably
forced to take into account on all possible states, oscillations. If a molecule
is in a state , then one expresses all possible states of the molecule byiteration of as:, $k\in\{0,1,2, \ldots\}$ .
$\mathrm{e}$

$x$

$f$

$f^{\lambda}.(x)$be a function. In order to express a large scale: $[0, 1]$Let
dynamics, one will consider the following family of the equations:$-\neq \mathrm{R}$$h$$\dot{x}^{\mathrm{A}}(t)\equiv-h(x^{\lambda}.)$,$x^{\lambda}.(0)=f^{j_{1}\backslash }(x)$represents a variable. We will call the above equation as feedback
whereequation for the oscillation. Iteration of the families of solutions:’$x^{\lambda}$.

$\{f^{l}(x^{\Lambda}’)(t)\}_{l_{\backslash }k}$will represent dynamics of the nlolecule, Let us put
as the solution states.
We will call$y_{l}.\cdot(t)=\{f’(x^{\lambda}.(t)\}f\cdot$$y_{\lambda}..(t)$2.B Simple patterns: Let us consider dynamics of the above equations.
and put $x\pm=x\pm\delta$ .
Suppose x is a fixed point by f, and take a small
Let us consider two cases that its derivative satisfies (a) $f’(x)>1$ or (b)are stable points.
$<1$ . For (a), bothare unstable points, and for (b),will be away froin x for (a), and approach to
Namely as kfor (b).
t$\delta>0$$x_{\pm}$$x_{\pm}$$arrow\infty,$$x$$f^{\lambda}|,(x_{\pm})$$\mathfrak{l})$$(\alpha)$

There are two cases;(1)$h(x)$$\geq 0$

or

(2)

A(x)

$\leq 0$.

Let us consider the case (a). After a small time, will move to x-forwill
for (2). In both cases (1) and (2), their iterations(1) andmove away from . Since the effect of $-h(x)$ also helps to move on thesame direction, These cases of dynamics will be strongly unstable. In thesecases the molecule will move to another states.
$x$$f^{\lambda}’(x_{\pm})$$x_{+}$$x$

4
as above. Notice that theSuppose (b) holds. Then will move to$karrow\infty$
will both converge to as. If (1) holds, then theiterations
effect of the dynam ics will try to make x-to move away from , so the
dynamics will be in an equilibrium state. If (2) holds, the dynamics will
help to converge to , and so it will be strongly stable. When two molecules
are interacting in strongly stable states, th en they will consist of a polymer .
For the case (b) and (2), we will say that self-feedback works.
We will gather these four patterns for the feed-back equation case in thefollowmoves to:$x$$x_{\pm}$$f^{\lambda}.(x_{\pm})$

$x$

$x$

$x$

$\cdot$$\mathrm{i}\mathrm{n}\mathrm{g};x$

(A)
(B)

$x_{-}$$x_{+}$(strongly unstable),$f’(x)>1$ ,$h(x)\underline{>}0$,(strongly unstable),$f’(x)>1$ ,$h(x)\underline{<}0$,(equilibrium state),$f’(x)<1$ ,$h(x)\geq 0$,(strongly stable),$f’(x)<1$ ,$h(x)\leq 0$

.

$\{$

(C)
(D)

$x_{-}$$x_{+}$(2.1)$\mathrm{D}$Next let us consider the case that is a periodic point, rather than afixed one. For simplicity, we will assume that the period is 2, $x<f(x)$and there are no other periodic points in $(x, f(x))$ .
At the beginning of time, the molecule is oscillating in $\{x, f(x)\}$ . Unlike
to the case of fixed points, here one needs to consider local behaviours of
iteration both at and $f(x)$ . In this case, we have two differntial equations:
$x$

$x$

.

$\mathrm{u}(t)\equiv-h(x^{0})$,$x^{(\}}(0)=x$,$\dot{x}^{1}(t)\equiv-h(x^{[perp]})$,$x^{1}(\mathrm{O})=f(x)$.

After a small time, the molecule also can choose any states.
Molecular oscillation is quite random, and so within a small time, it willtake any states in.
If the case (C) hold at both and $f(x)$ , then the self-feedback does workfor bothand , and so the molecular states will stay near $\{x, f(x)\}$ .
If (1) holds at both and $f(x)$ , then its states cannot stay near $\{x, f(x)\}$ ,and they will move near another periodic points alongor
randomly. There are another cases, likestay nea.r andwill move to another states (for example (C) at and (1) at $f(x)$ , etc).
$\{x^{1\mathrm{J}}(t), x^{1}(t)\}$$\{f^{\lambda}.(x^{0}(t)), f^{\lambda}.(x^{1}(t))\}_{\lambda}.$$x$$x^{0}$$x^{1}$$x$$f^{\lambda^{\iota}}.(x^{(\}}(t))$$f^{l}’.(x^{0}(t))$

$x$

$x$

$f^{\lambda\prime}’(x^{1}(t))$$f^{\mathit{1},\prime}’(x^{1}(t))$

5
In these cases, after a small time the states lnay stay near , or move toanoter randomly.
We will say that self-feedback works if each solution state $y:(t),$ $i=0,1$
stays near a periodic orbit. In that case the ensyme takes its states near
the periodic orbit as the next time step.
We have described very simple patterns, but in general the orbits ofstates will be much more conlplicated, and may show choastic behaviour.
$x$

3Interaction

3.A Simple interactions: In our framework, a state of an ensyme
is expressed by an element x $\in[0,$ 1] near $P(n)$ for some 7b. Now let us$A$

as a reptessor, and suppose . acts onandand(activator resp.). Suppose that at a timne t these are in states
respectively. Repressing (resp. activating) dynamics are expressed bydifferential equations as:

talqe two proteins$A_{1}$$A_{2}$$A_{1}$$A_{\underline{7}}$$x_{1}$$x_{2}$$\dot{x}_{1}^{\mathrm{A}\}}(Jt)=-h(x_{2}^{l\mathrm{c}’})$(repressor) or,(activator).

$\dot{x}_{1}^{\lambda}.\cdot(t)=h(x_{2}^{\lambda})$Combinig with self-beedback, one can totally write the equations for theactivator case as:$\dot{x}_{\acute{1}}^{l}\acute{.}(t)=h(x_{\sim^{)}}^{\lambda}‘.)-l\iota(x_{\grave{\acute{1}}}^{\mathit{1}})$,$x^{l}\mathrm{i}.(0)=f^{l}’.(x_{1})$,$x_{)}^{t,\prime}\zeta.(0)=f^{\mathrm{A}}.(x_{2})$

.

$\{$

$\dot{x}_{2}^{\lambda}.(t,)=-h(x_{\underline{\eta}}^{\mathrm{A}}.)$

,

(3.1)

Let us consider the patterns of this dynamics as in the previous section. In, and for the caseonthis simple case there are no interaction fromits behaviour as in the section 2. Let us take a fixed
one canof. There are two cases:
point . The point is the sign of$A_{1}$$A_{2}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$x_{2\mathrm{J}}$$h(x_{\gamma}^{\bigwedge_{\sim}}.\cdot)-h(x_{1}^{\mathrm{A}^{\acute{\lambda}}})$$x_{1}$(1)$l\iota(x_{2}^{l}’)-h(x_{1}^{l,}.)\geq 0$,(2)$h(x_{2}^{h}.)-h(x_{1}^{\lambda}.)\underline{<}0$.

If the molecule starts from , then one may follow the dynarnics ofby the same way as section 2. Let us put $x_{\pm}=x_{1}\pm\delta$ , and take anotherhas the state . Then we$x_{1}<x_{1}’<x_{+}$ . suppose at the beggining,moves to:have the following diagram;$A_{1}$$\prime x_{1}$$A_{1}$$x_{1}^{r}$$x_{1}’$$x_{+}$$x_{1}’$$\{$$x_{1}’$$x_{1}$(strongly unstable) ,$f’(x_{1})>1$ ,$h(x_{2}^{\lambda})-h(x_{1}^{\lambda}.)\geq 0$,(equilibrium state),(equilibrium state) ,$f^{l}(x_{1})>1$,$h(x_{\underline{)}}^{\lambda}.\cdot)-h(x_{\acute{1}}^{1}.)\leq 0$,$f’(x_{1})<1$ ,$h(x_{2}^{h})-h(x_{1}^{h}.)\geq 0$,(strongly stable),$f’(x_{1})<1$ ,$h(x_{2}^{l}’)-h(x_{1}^{h}’)\leq 0$

.

(3.2)

$\epsilon$.
For example suppose $f’(x_{1})>$ I and bothandarenegative. Then as the quantity ofis gradually increasing, the state ofcannot stay near . Namely
will change from an equilibrium state
into a strongly unstable one.
One may consider the case whenstarts at the state $x_{-}<x_{1}^{tt}<x_{1}$ bya) parallel way.
Letandbe the solution states respectively. If
stay someperiodic orbits
respectively, then we will say that this system isandstable. Also in that case these ensymes take states nearandrespectively as the next time steps.
$h(x_{2}^{l}.)|$$h(x_{\underline{9}}^{\lambda^{\iota}})-h(x_{1}^{\lambda}$$)$$A_{2}$$A_{1}$$A_{1}$$x_{1}$$A_{1}$$y^{1}(t)$$y\underline’(t)$$p_{1}$$y^{r}(t)$$p_{2}$$p_{1}$$p_{2}$

3.B Intermediate time scale of interactions: Let us consider systems
between ensymes
andwhich interact mutually as activators. Taking
into account of self-feedback, one obtains an activating system on the large
scale of time as:$A_{1}$$A_{2}$$\dot{x}_{1}^{\mathrm{A}}.(t)=h(x_{2}^{\mathit{1},J}.)-h(x_{1}^{\lambda i})$,$x_{1}^{\Lambda\backslash }(0)=f^{h_{i}}(x_{1})$,$\dot{x}_{2}^{l\iota}.(t)=h(x_{1}^{\lambda}.)-h(x_{2}^{l}’.)$,$x_{2}^{l}’.(0)=f^{\mathrm{t}_{j}}(x_{2})$

.

(3.3)

$\{$

By solving these equations, one obtains two sets of solutions
and.
If one takes into account of the order to interact, then one needs toconsider smaller time scale. For example firstacts on , and after theeffect of the interaction
becomes an activating state and then is able toact onconversely, and so on.
There are two different initial directions of interactions:$\{x_{1}^{\lambda}.\}$$A_{1}$$\{x_{\underline{?}}^{l_{1}}.\}$$A_{J,\lrcorner}.$$A_{2}$$A_{1}$$A_{1}arrow A_{2}$,$A_{2}arrow A_{1}$.

For the first case, one starts fromand it interacts. Thenacts on. They repeat to interact mutually. For the other case, onechanges the roles ofand. Let us consider the first case, and follow
its procedure. First one solves the equation:$A_{1}$$\mathrm{o}\mathrm{n}^{-}[perp] A_{2}$$A_{1}$$A_{1}$$A_{-},$.
$\dot{x}_{1}^{\lambda}$

Then using$\{x_{1}^{t_{i}}\}$$=-h(x_{1}^{l\iota}.)$,$x_{\mathrm{I}}^{\lambda}.(0)=f^{h}(x_{1})$.

, one solves another equations:$\dot{x}_{2}^{\lambda}.(t)=h(x_{1}^{\prime_{\tau}}.)-h(x_{2}^{l_{1}}.)$,$x^{k}?.(arrow 0)=f^{\lambda}(x_{2}.)$.

One considers another equations:,$\dot{x}_{1_{\backslash }2}^{\lambda}.(t)=h(x_{\underline{9}}^{\lambda^{\mathfrak{l}}})-h(x_{1_{\backslash }2}^{\lambda_{i}})$$x_{1.2}^{\lambda}.(0)=f^{\lambda\prime}.(x_{\mathrm{J}} )$.

$A_{2}$

7
One iterates this process. Namely consider the equations:$\dot{x}_{2_{\backslash }2}^{\lambda}.(t)=h(x_{1.2}^{l}’.)-h(x_{92}^{\lambda}.\cdot.)$

Using$\{x_{2_{\backslash }2}^{\lambda^{\backslash }}\}$,$x_{2_{\backslash }\underline{7}}^{\mathit{1}^{\mathrm{Y}}}.(0)=f^{lt}.(x_{2})$.

, again one solves the followings:,$\dot{x}_{1_{\backslash }3}^{\lambda}.(t)=h(x_{\underline{?}2\backslash }^{l\mathrm{t}}.)-h(x_{1.3}^{\lambda}.)$$x_{1.3}^{\lambda}.(0)=f^{\lambda^{\tau}}(0)$.

Succesively iterating to solve the equations, one obtains families:,$\{x_{1_{\backslash }r\}\}}^{\lambda_{i}}\}$$\{x_{2_{\wedge}?7\mathfrak{l}}^{\lambda}..\}$

.

,
and
Let us recall that in the first paragraph of $3.B$ , we havefamilies of solutions in the large scaling time. Now we will say that thescale, if for eachoriginal interaction system is stable in smaller
stay near the same periodic orbit for all, andand $\mathrm{i}=1,27$ both. forsufficiently large . We will clenote the situation bylarge .
, then thetoLet us consider the converse path. If one starts from. We will say thatparallel argument will yield another families
is cyclic, if:andthe original dynamical system with the intial states$\{x_{1}^{\lambda}\acute{.}\}$$\{x_{\underline{9}}^{l}’.\}$$k$$t\mathrm{i}7ne$$x_{7_{J}\backslash 7\}}^{\lambda}.$$x_{?}^{\lambda}.\cdot$$\{x_{7_{\backslash }?}^{\lambda},,\}_{\lambda_{i}}\sim\{x^{l}\mathrm{i}.\}_{l_{1}}$

$?n$

$m,$

$A_{2}$$\{w_{1_{\backslash },,?}^{\lambda}.

, w_{\underline{?}_{7’ \mathrm{I}}}^{1\supset}\backslash \}$$x_{2}$$x_{1}$$\{x_{1_{\mathrm{t}}7l1}^{l,}.\}_{l\iota}\cdot\sim\{w^{l}\mathrm{i}_{?1\}}.\backslash \}_{\lambda_{i}}$,$A_{1}$.

$\{x_{1_{\backslash }?t\prime}^{\lambda}.\}_{\lambda}\cdot\sim\{_{l\mathrm{L}_{1_{\backslash }?\prime)}’}^{j_{\mathfrak{i}}}’.\}_{\lambda}$

hold for all sufficiently large $m$ .
This is the simplest circuit of the interaction system. A little bit moregeneral circuit will be as:$A_{1}arrow A_{2}arrow.$..

$A_{\mathit{1}\mathrm{V}}arrow A_{1}$.

One formulates stability for this system completely by the same way. One, and consider the follow ing path:
may start from
$A_{2}$$A_{2}arrow A_{3}arrow.$. . $A_{N}arrow A_{1}arrow A_{2}$ .

In this case also one obtains a notion of cyclicity.
4

Systems

Let us consider a system by mutually interacting ensymes:$\{A_{1}, \ldots, A_{N}\}$

respectively. One can consider various
with the initial statesroutes of interaction. In fact, for each 0riented signed graph with self-loops$\{x_{1}, \ldots, x_{N}\}$8and with a subset ofas vertices, there associated with aninteracting system. For example for a full graph (every pair of edges isconnected), one obtains a family of the equations:$\{x_{1}, \ldots, \prime x_{N}\}$$\dot{x}_{1}^{tJ}.(t)=-h(f^{\lambda}.(x_{1}))+\Sigma_{j=}^{N}-,\pm h(x_{j}^{\lambda}.)$,$x_{1}^{l\mathrm{t}}(0)=f^{\lambda}.(x_{1})$,$\dot{x}_{\underline{?}}^{\mathrm{A}\prime}.(t)=-h(f^{j}’.(x_{2}))+\Sigma_{j}\neq 2\pm h(x_{j}^{\mathrm{A}}.)$,$x_{2}^{\lambda i}’(0)=f^{L_{i}}(x_{2})$

,

$\{$

.

...,

$\dot{x}_{N}^{\mathrm{A}}$$(t)=-h(f^{\lambda}.(x_{N}.))+\Sigma_{J}\neq N\pm h(x_{j}^{\mathit{1}}’.)$,$x_{N}^{\lambda}(0)=f^{h}’.(x_{N})$

.

(4.1)

Let
be an oriented signed graphset of vertices is consisted bya subset of. We will say that the set
is apreinteraction system, and $(f$ ; , . . . ,; $G)$ is an interaction systern. For an
oriented signed graph, we will say it is an interaction graph.
Let $(f; x_{1}, \ldots, x_{N}; G)$ be a stable interaction system in the sense of 3. .
We will say that it is stable under small perturbatiori, if there exists
such that for any
,...,
with
, another interaction system$G)$$(f$ ;,. .,;is also stable.
$G$$\mathrm{w}1_{1}\mathrm{o}\mathrm{s}\mathrm{e}$$\{x_{1}, .

.

.

$(f; \prime x_{1}, \ldots, x_{N})$, x_{N}\}$$x_{N}$$x_{1}$$A$$\epsilon>0$$x_{N}’)$$(x_{1}’$,$x_{1}’$$|x_{j}-x_{\iota}’.|<\epsilon$$x_{N}’$

4.B Cyclic system: Let us consider a system consisted by N ensymes$A_{N})$ .
Suppose eachis being activated by, and catalyzes, where we consider mod N. Eachontakes a state
whensilence,being
andwhen activating.
For example suppose there exists a pairfor each i suchthat bothandare fixed points, there are no other periodic points
betw een them and $f(x)<x$ hold for x.
We take. The dynamics can be written as:$(A_{1},$$A_{7}-1$$A_{j}$\ldots ,,$A_{\tau+1}$$A_{i}$$x_{+}(\mathrm{i})$$z_{-}(\mathrm{i})$$x(\mathrm{i})<\sim\sim(\mathrm{i})$$x(\mathrm{i})$$\sim\sim(\mathrm{i})$$\in(x^{\tau}(\mathrm{i}), \sim(\sim \mathrm{i}))$$x(\mathrm{i})<x_{+}(\mathrm{i})<<\approx_{-}(\mathrm{i})<\approx(\mathrm{i})$$\dot{x}$Thus whenj.

$=-x_{7}^{\lambda}$.
$+x_{j-1}^{\lambda}\backslash$.

is in a low state, tlen
will also be in a low state andnear. If the state ofis high and near1), then the stateofwill be also high and near.
This is a stable interaction system,.

$A_{j-1}$$A_{l}$$A_{j-1}$$x_{+}(\mathrm{i})$$A_{i}$$z_{-}(\mathrm{i}$–$z_{-}(i)$4.C Cirrgadian rythms: Let G be an oriented signed graph. We will
say that G lifts periodically, if an infinite numbers of G can be connectedperiodically. We will denote the result by .
Let $(f;x_{1},$ $x_{N};$G) be a stable interaction system which lifts to alsoa stable periodic system (f;...,.
We will say that , is a clock gene, if any of the interaction systems as$(f:x_{1},$$x_{N};G’)$ cannot lift stably.
$x_{7??-1}, x_{r?l+1},$$\tilde{G}$\ldots ,$x_{1},$$x_{7?}$\ldots ,\ldots ,$x_{N};\tilde{G})$a
Evolution5

Let $(f;x_{1}, \ldots, x_{N})$ be a pre-interaction system, and$\mathfrak{P}=\mathfrak{P}(f;x_{1}, .

.

.

, x_{N})=\{P_{1}$,...,$P_{\mathrm{A}}.\}$

be a set of stable interaction graphs. Vertices of are consisted by subsets. We will say that the set is stable unler small perturbation,ofwith $|xj-x_{j}’|<\epsilon$ , another,...,such that for any
if there exists
admits also the same set of the stablepre-interaction systeminteraction graphs.
Recall that the colon corresponlence is not one to one. For each aminoacid, there associated with several codon triples. If a mutation does notchange the corresponding amino acid, then it would not influence on the
mutacorresponding protein itself. When this type of mutation (calledtion) has occured, the interaction system itself will not be influenced. It is
known that this type of mutation occurs very often in a real life.
We will say tha.t an interaction system $(f$ ; , . . . , ; $G’)$ is an apparent
mutant of $(f; x_{1}, \ldots, x_{N};G)$ , if the corresponding sets of intaraction graphs
are the same, $G=G’$ .
Let us regard that a set of interaction graphs repersents a macroscopic
be two sets of interaction graphs withandsystem of a life. Let
respect to $(f; x_{1}, \ldots, x_{N})$ and $(f; y_{1}, \ldots, y_{N})$ respectively. We will say that, the following
are mutually close spicies, if for any small
andwhich is an
hdds; there exists an interaction system, and another interaction systemapparent mutant to”is, such thatwith.
an apparent mutant to$P_{i}$$\{x_{1}, \ldots, x_{N}\}$$\epsilon>0$$x_{N}’)$$(x_{1}’$$(f\}.x_{1}’ , .

.

.

, x_{\mathit{1}\mathrm{V}}’)$$x_{N}^{l}$$x_{1}’$$\mathfrak{P}_{1}$$\mathfrak{P}_{2}$$\epsilon>0$$\mathfrak{P}_{2}$$\mathfrak{P}_{1}$$(f; x_{1}’, \ldots, x_{N}’; \mathfrak{P}1)$$(f; x_{1}, \ldots, x_{N};\mathfrak{P}1)$$(f; x_{1}", \ldots, x_{N}" ; \mathfrak{P}_{2})$$|<\epsilon$$|x_{i}’-x_{\dot{7}}$$(f:x_{1}", \ldots, x_{N}" ; \mathfrak{P}_{2})$$(f,\cdot y_{1}, \ldots, y_{N}; \mathfrak{P}_{2})$and consider to obtain
and
Example: Let us consider two ensymes
equilibrium systems. As before we consider a simple example. Suppose
has two fixed points $x>x’$ , there are no other period ic points between
them and $s>f(s)$ hold for all $s\in(x’, x)$ . Let us consider the situation
as an enhanser. Take $x_{+}=x+\delta$ , and consider a preacts on
thatinteraction system $(f;x_{+}., x_{2})$ where may be arbitrary. In order to be inan equilibrium state, A2 catalyzes as a repressor (section 2):$A_{1}$$A_{2}$$f$$A_{2}$$A_{1}$$x;2$$\dot{x}_{1}^{\mathrm{A}j}(t)=-f^{\lambda}.(x_{1})-x_{2}^{\lambda^{\backslash }}$,$x_{1}^{\lambda}.(0)=f^{\mathrm{A}}.(x_{+})$$\{$$\dot{x}_{2}^{\lambda j}(t)=-f^{\lambda j}(x_{2})$,$x^{\mathrm{A}\prime}\underline,(0)=f^{\lambda i}(x_{2})$

.

,

(5.1)

10
has become toand suppose
Now suppose a mutation occurs on
plays a role of repressor, the states
take its state as $x_{-}=x-\delta$ . As far as
will fall down into . After such process, if one wants to obtain anofshould catalyze as an activator.
equilibrium state near $x’,$
andchanges.
In this case the sign of the edge between$A_{1}$$A_{1}$$A_{\underline{?}}$$x^{\mathit{1}}$$A_{1}$$A_{2}$$A_{1}$$A_{2}$

5.B Evolution dynamics by enhansers: In some occasions, macroscopic evolution has occured by mutation of enhansers. This shows th atsome macro phenomena can be understood by the strength of the activityof molecules.
a simple pre-interaction system,
be smali, and let us
Let
and two dynam ics:(f;$\epsilon>0$$\mathrm{t}\mathrm{a}\mathfrak{l}\mathrm{k}^{\vee}\mathrm{e}$$x_{1}, x_{2})$,$\dot{x}_{1}^{h}.(\mathrm{f})=-h(x_{1}^{l}.)\pm h(x_{2}^{\lambda_{i}})$$x_{\acute{1}}’*\cdot‘(0)=f^{\mathrm{A}}.(x_{1})$,$x_{2}^{\lambda}.(\mathrm{O})=f^{l_{i}}’(x_{2})$

.

(5.2)

$\{$

$\dot{x}_{2}^{\lambda^{\backslash },}(t)=-h(x_{2}^{\mathrm{A}!})$,$\dot{x}_{1}^{\lambda^{\acute{1}}}(t)=-h(x_{1}^{\lambda}.)\pm(1+\epsilon)h(x_{?,\sim}^{l1}.)$,$x_{1}^{\Lambda}.(0)=f^{J\mathrm{t}}.(x_{1})$,$x_{2}^{\mathrm{A}}(0)=f^{\lambda}.(x_{\mathrm{A}}’)$

.

$\{$

$\dot{x}_{2}^{h},(t)=-h(x_{2}^{\lambda 1})$

,

(5.3)

.

acts onas an enhanser, but its strength are mutually
In both cases,. be the set of solutions for the second dynamics
different. Let
and changes it to . Then one(5.3). Suppose a mutation occurs on
again considers the dynamics:$x_{1}^{\lambda j}$$x_{2}^{\lambda}$$\{x_{1}^{\lambda i}, x_{2}^{\lambda}.\}_{\lambda}$$x_{3}$$x_{2}$$\dot{x}_{1}^{k^{1}}(t)=-h(x_{1}^{\lambda})\pm h(x_{3}^{\lambda^{1}})$,$x_{1}^{h\prime}.(0)=f^{\lambda}.(x_{1})$,$x_{3}^{l}.(0)=f^{\lambda}.(x_{3})$

.

(5.4)

$\{$

$\dot{x}_{3}^{\lambda}(t)=-,h(x_{3}^{h})$,Letbe the set of the solutions for this dynamics. We will say thatthe evolution is induced by a mutation of enhartsers, if the followings hold
( $3.B$ for ) :.
,,$\{z_{1}^{\mathrm{A}}., \sim \mathrm{Y}\}_{t;}3\lambda$$\sim$$\{x_{1}^{\lambda j}\}_{l_{i}},\sim\{_{\sim}^{\lambda_{i}}\vee\}_{\lambda}1’$$\{x_{2}^{h}.\}_{\lambda\prime}.\sim\{_{\sim}^{\lambda}\sim.\}_{\lambda}3,,$, andExample: Let us choose two ensymes as in the example inacts onconsider the situation thatas an activator. Let us consider
fixedagain to obtain equilibrium systems. In this case suppose haspoints $x<x’$ , there are no other periodic points between them and $s<f(s)$hold for all $s\in(x, x’)$ . Take $x_{+}=x+\delta$ , and consider an interaction system$(f;x_{+}, x_{2})$ :$\mathit{0}^{\mathrm{r}}.A$$A_{2}$$A_{1}$$f$$\dot{x}_{1}^{\mathrm{A}}.(t)=-x_{1}^{\mathit{1}\iota\acute’}+x_{\dot{2}}^{l}.$,$x_{1}^{\lambda}.(0)=f^{h}.(x_{+})$$\{$$\dot{x}_{2}^{\lambda}\acute{.}(t)=-x_{2}^{\lambda_{i}}$,$x_{2}^{\lambda}\acute{.}(0)=f^{L^{1}}(x_{2})$,$\mathrm{t}\backslash \mathrm{v}\mathrm{o}$

(5.5)

11
.

In order for this system to be in an equilibrium state, the inequality
should hold.
Supposemore strongly. Let us change thebegins activating on:
notation of the variable$x_{1}^{\lambda}$$>x_{\dot{2}}^{lj}$$A_{1}$$A_{2}$$\mathrm{b}\mathrm{y}\approx_{1’}^{\lambda}.$.

$\sim.1(\sim^{\lambda_{i}}t)=-^{\lambda}\sim 1^{\cdot}+\sim(1+\epsilon)x^{\lambda}\underline,\cdot>0$

will increase
When the sign of the right band side has changed, thenand approach near . After some time, this will become an equilibrium.
.
will increase until nearstate, since the value$\approx_{1}^{\lambda J}\backslash$$\prime x’$$(1+\epsilon)x_{2}^{\lambda^{1}}$$z_{1}^{\lambda}$

5.C Conservation of structure of proteins: If homology of two sequences of genes are more than 30 percent, then one may guess these will
share with the same ancient, and will have very similar protein structures
or functions.
Some genes have mutually very different sequences, but the corresponding proteins have turned out to be very resemble each other. This wasfound by X-ray analysis, e.g., miogrobin and hemogrobin for some spicies.
In an interaction network, functions of proteins will tend to be conserved,
but within such conservation, sequences might be changeable frequently.
. Then we put:Let us choose a family of periodic points$\{p_{1}, \ldots,p_{N}\}$$P(\{p_{1},$$p_{N}\})=\{G:(f;x_{1},$ \ldots ,$x_{N};$ G)\ldots ,:the solution states$\{f^{l}(x_{7}^{\lambda}.)\}_{\lambda_{\backslash }}.,$,

stay near$p_{i}$}.

Bio-functional quantity

6

Liapunov exponent: Let$x\in[0,1]$ , the Liapunov exponent$6.\mathrm{A}$$f$: $[0, 1]\vdasharrow[0,1]$ be ais defined by:$G^{1}$

map. Then at$\lambda(x)$$\lambda(x)=\lim\sup,\frac{1}{n}\log(|(f^{?l})’(x)|)$$= \lim\sup_{r\acute,arrow \mathrm{x}}\frac{1}{n}\Sigma_{j=\mathrm{U}}^{11-1}.\log(|f’(f^{j}(x))|)$.

When this number is negative, then is stable. Namely A satisfies somne
continuity in some sense, and any orbits near will approach to the oneof . If $\lambda(x)>0$ , then the situation will drastically change. It will be
far from to be continuous, and orbits will be sensitivel influenced by the
initial points near . In fact one has the Taylor expansion:
$x$

$x$

$x$

$\mathrm{y}$$x$$|f^{7l}(x+\delta)-f^{\tau 7}(x)|\sim|(f^{?1})’(x)\delta|\sim|\delta|L(x)^{\tau l}$,$L(x)=\exp(\lambda(x))$ .

12$L(x)$

satisfies $1>$ or $<1$ according to the signs of$\lambda(x)$.

6.B Simple cases: Let us consider the simplest case of interaction system,and try to obtain its-functional qttantity.
Let (f;x) be the single feed back system which is follow ed by the dynam ics:$b\mathrm{i}o$$\dot{x}^{l,\prime}(t)=-x^{\lambda}.$Then one defines a parametrized,$x^{l}’.(0)=f^{k}(x)$$\mathrm{b}\mathrm{i}\mathrm{o}$.

-functional quantity by:$\lambda(\alpha, x)=\lim\sup_{\gamma \mathrm{p}arrow\infty}\frac{1}{n}[\log(|(f^{t1})’(x)|)+\alpha\log f^{\uparrow 1}(x)]$,$\alpha>0$.

In general the inequality holds,.
One may have a picture for this ixumber as follows. Let x be a fixed
point of f.
$\lambda(\alpha, x)\leq\lambda(x)$(1) If $|f’(x)|>1$ , then:(a) $x_{-}=x-\delta$ will be strongly unstable with respect to(b) $x_{+}=x+\delta$ will be an equilibrium point,$x$,(2) If $f’(x)<1$ , then:(c)will be an equilibrium point,(d) $x_{+}=x+\delta$ will be strongly stable with respect to .
$x_{-}=x-\tilde{\delta}$

$x$

(1)

$\mathrm{t}^{L}l$

The above suituation is expressed by theA(J)We will say that$<\lambda(c)$-functional quantitiy:$\mathrm{b}\mathrm{i}\mathrm{o}$, A(b) $<$ A(a).

x) in (b) or (c) are equilibrium invariants.

$\lambda(\alpha,$

6.C Interaction system by two ensymes: For interaction systems
consisted by more than two ensymes, one can obtain also bio-functional
quantities. For sim plicity, suppose h is a map betw een intervals h : [0,[0,1], and consider a system with feedback plus enhanser:$1]\vdash+$$\dot{x}_{1}^{\lambda}.(t)=\pm h(x_{2}^{\mathrm{A}}.)-h(x_{1}^{\lambda_{i}})$,$x_{1}^{\lambda}’(0)=f^{\lambda}(x_{1})$,$x_{2}^{\lambda j}(0)=f^{\mathrm{A}}.(x_{2})$

.

$\{$

$\dot{x}_{2}^{\lambda^{1}}(t)=-h(x_{2}^{\lambda^{\acute{1}}})$

,

(6.1)

13
In this case one obtains parametrizedA-functional quantittes by:$\mathrm{b}\mathrm{i}\mathrm{o}$$(\alpha, x_{1})=$$\lim\sup_{77arrow\infty}\frac{1}{n}[\log(|(f^{f1}.)’(x_{1})|)-\alpha(\pm\log h(f^{?7}(x_{\underline{9}}))-\log h(f^{?t}’(x_{1})))]$$\lambda(\beta, x_{2})=\lim\sup_{r\iotaarrow \mathrm{X}}\frac{1}{n}[\log(|(f^{r\lambda})’(x_{2})|)+\beta\log h(f^{;\}}(x_{2}))]$For more general stable systems
functional invariants:(A$(\alpha_{1,.1}\prime x\cdot),$$(f; x_{1}, \ldots, x_{N}; C_{\tau})$$\lambda(\alpha_{N},$$\ldots,$$x_{N})$,.

, one obtains tie bio-)by the completely analogous way.

Iterating functions with parameter7: $[0, 1]-+[0,1]$ be a family of maps. When the Schwartz derivatives
Letof , are all negative, like the Logistic maps, then the behaviour of periodic. Then as thepoints are well studied. Suppose is a fixed point oflooks like as:parameter moves from to $s_{1}=s_{(1}+\delta,$$f_{\epsilon}$$f_{2}$$x$$f_{h_{\{1}}’’$$f_{\epsilon}$$s$$s_{0}$$\{|\}$–p$\zeta^{-}Lj$–\rangle

For (1), let us consider the single feedback system in $6.B$ . During the
movement of the parameter, $x_{-}=x-\delta$ drastically falls down into another. $x_{+}=x+\delta$ is strongly stable at , and changes into an
periodic point
equilibrium state at .
$x_{-}^{1}$$s_{\langle\}}$$s_{1}$: $[0, 1]arrow[0,1]$ be a param etrized map, andDoubling genes: Let
suppose periodic orbits have pitchfork bifurcation, e.g., the logistic map$f_{h}$$9.\mathrm{B}_{\backslash }$$f_{\mathit{8}}(x)=sx(1-x)$.

In neutral theory in evolution, bifurcation of spicies is explained by mutation by doubling genes. If a doubling of a gene occurs, then in orderto keep the system, it will be enough if one of them plays the role of theoriginal function. So the other may be free to mutate if the result would
not affect the system itself.

14
In our setting, one may have a picture of doubling genes by regarding
as the evolutional time parameter.
Let us take an embedding of two oriented signed graphs $G\subset H$ . Let
be an interaction system, and let us move the parameter. At the point $s^{1}=s+\delta$ of bifurcation, let us double some states, say :$s$$(f_{s};x_{1}, .

.

.

, x_{N};G)$

$x:$

$s$

$(f_{b}1;x_{1}, x_{2}, \ldots, x_{?-1}, x_{i}, x_{i}, x;+1, x_{i+2}, \ldots, x_{N};G)$is in. Supposecorresponds to the first ofwhere a vertex ofan equilibrium state before bifurcation. One may perturbe the otherslightly atso that it will move to another periodic orbit.
$G$

$Xj$

$x_{j}’ \mathrm{s}$$Xj$$s^{1}$As a result one will obtain another interaction system at$(f_{b}1 ; x_{1}, , x_{2}+’ .

.

.

, x_{\dot{7}-1}, x_{j}, x_{i’+1}’x,, x_{7+2}, .

.

.

$s_{+}^{1}=s^{1}+\delta$:, x_{N}; H)$

is consisted by .
where a vertex of.
’isbe the ordered set of bifurcation points.
Let $s=s^{0},$known as the Feigenbaam constant for the logistic map. At each time ,one may choose an evolutionized interaction system:$H\backslash G$$x_{\dot{7}}’$$1\mathrm{i}_{\mathrm{l}}\mathrm{n}s$$s^{1},$$=s^{\mathrm{x}}$$\ldots$$\tau 5^{\tau}:$$n$$3_{\iota}=(f_{\dagger^{j}}, ; x_{1}^{\dot{7}}, \ldots, x_{i_{N}}^{7} ; H_{i})$,$H_{\{)}=G\subset H_{1}\subset\cdots\subset H_{j}\subset\ldots$

The family

8$\{3j\}_{7}$represents an evolution of a system by doubling genes.

Switching of genes

Ensymes play roles of switching in a biological systems. Let us choose an
ensyme in a system. The system contains two states, namely when theensyme is activating and catalyzing the other molecules, and another iswhen being silence. Let us represent this situation in our setting.
Let $(A_{1}, \ldots, A_{N})$ be the set of molecules which consists of an interaction
system. Whenis being silence, and in a state , its interaction system
may be expressed by:$A_{N}$$\approx$$(f;x_{1}, .

where$C_{7}$

does not contain$\approx$.

.

, x_{N}, \approx;G)$

as a vertex.

15
Suppose $A_{N-1}$ works as a -ensyme on
Then the system will change as:$\mathrm{c}\mathrm{o}$$(f;x_{1}^{l}, .

.

.

and

$A_{N}$$A_{N}$

begins activating.

, x_{N}’’, "\sim;H)$

where $H$ contains and may be very different from .
A simple description of switching was given in $3.A$ .
$G$

$z’$

be a system
Systems by binary enhansers: Let: . . . ,is catalyzed by another ensyme
consisted by ensymes.
$\{A_{1,)}\ldots A_{N}\}$ , and it takes two statesaccording to whether itoror not. Suppose each , catalyzes another two ensymes
is catalyzed byis in states or respectively. Namely
or $C_{?}\in\{A_{1}, \ldots, A_{N}\}$ whenchanges the partoners to calalyze whether it is activating or not. Then
is a brnavy enhanser and works as a switching.
we will say that
$A_{N})$$(A_{1}$$8.\mathrm{B}$$A_{i}^{t}\in$$\mathrm{E}\mathrm{a}_{1}\mathrm{c}\mathrm{h}A_{:}$$x_{i}$$\sim:\sim$$A$$A_{i}’$$A_{j}$$B_{i}$$x_{7}$$z_{j}$$A_{i}$$A_{j}$$\mathrm{B}t\backslash$$/^{\eta}\lambda_{\grave{l}}$

A

$t’/$

$arrow$$\mathrm{A}_{\mathrm{I}}^{\backslash }$$\mathrm{j}\mathrm{Z}_{\grave{1}}^{[searrow]}$$\mathrm{C}\mathrm{I}\backslash$takes, and suppose
Example: Let us consider a system by
two states between the interval $(x(2), \approx(2))$ where $x(2)$ and $z(2)$ are both
fixed points, there are no other periodic points and $f(x\cdot)<x$ hold in the
takes two states between $(x(3), \approx(3))$ which has the same propinterval.
takes two states,
erties except that $f(x)$ $>x$ hold in the interval.
.
andsatisfying
andwhere may
Let us consider a pre-interaction system $(f; x, x_{+}(2),$and A3 admit the dynamics:
be arbitrary.
$(A_{1}:$A_{2}$

A_{2}, A_{3})$$A_{3}$$A_{1}$$h(x_{1})\backslash ’<0$$\approx_{1}$$x_{1}$$h(_{\sim 1)}^{\gamma}\backslash >>0$$x$$\sim.-\sim(3))$$A_{2}$$\dot{x}^{\lambda}.(\mathrm{i})=-x^{\lambda\prime}’(\mathrm{i})+h(x_{1}))$

When $x=x_{1}$ , thenon .

$A_{1}$

catalyzes only on$A_{2}$$\mathrm{i}=2,3$.

, and when$x=\sim 1\sim$, it does only

$A_{3}$II Interaction by composition of maps
Below we will show one realizat.ion of interaction of maps by way of conipositions which has described in part I. Under such formulation, we havestudied several analytic properties and relations with other fields in [K2].
to a larger
Passing through small scaling in iterations of maps
, flow of the interaction maps, we
one in another iteration
obtain a pattern formation from micro interactions.
$7\iota$$t$$\{\Phi(x)^{t}(\overline{k})\}_{f}$$\{l\iota^{t?}\}$

1$\mathrm{G}$9Interaction of maps

Let us take two interval maps:: $[0, 1]arrow[0,1]$$f,$ $g$

and consider their iterations:01$(X)=\{f^{\lambda}.(x)\}_{\lambda=\mathit{0},1,\ldots}$,$O_{2}(x)=\{g^{l}’.(x)\}_{\lambda\cdot=\mathit{0},1,\ldots}.$

We call them as the oscillatins ([K2]).
Let us have a symbolic dynamics
by two alphabets {0, 1}. Then foreach element $k=(k_{0}, k_{1}, \ldots)\in X_{2}$ , we will associate a family of maps:$X_{2}$$\{h^{\lambda}.(x)\}_{l_{\iota’}=0_{\backslash }1_{\backslash }}\ldots$

as follows. Let us put:$f(x)$$i=0$ ,

$g(x)$$i=1$ .

$d_{j}(x)=\{$

Then we define$h^{\lambda}$

.

by:
.

$h^{\mathrm{A}}.(x)\equiv d_{\lambda}$$\mathrm{o}d_{h\mathit{1}-1}\circ\cdots\circ d_{\mathrm{f}\mathrm{I}}(x)$

.

Let :
$\pi$

: $[0, 1]arrow\{0,1\}$be a measurable map. For example one may chooseand. Then for each $x\in[0,1]$ , one can compose the interaction
with . Thus one obtains another elem ent for. :$\tau_{1}([0,\frac{1}{\mathit{2}}))\equiv 0$$\pi((\frac{1}{2},1])\equiv 1$$\pi$$\mathrm{a}.\mathrm{e}$$x$$\pi((h^{1}(x), h^{1}(x),$ $\ldots))\equiv(\pi\circ h^{0}(x), \pi \mathrm{o}h^{1}(x),$ $\ldots)\in X_{2}$.

We call:$\Phi(x, f, g)$:by $\Phi(x, f, g)(\overline{k})\equiv\pi((h^{0}(x), h^{1}(x),$We call $x\in[0,1]$ a regulcvr point, if$J\mathrm{Y}_{2}-+J\mathrm{Y}_{2}$as the interaction map.
$\Phi(x)$ is a homeomorphism. Let:$\ldots))$$G=G(f, g)=$ { $\Phi(x)$ : $x\in[0,1]$ , regular}$\subset$Aut$z\mathrm{Y}\underline{\eta}$

be a group generated byfor all regular points .
One can generalize the construction of
by using more than two
maps. In fact using four interval maps $f,$ and, one can obtain the$G(f,g,\alpha,\beta)$
corresponding interaction map, and.
$\Phi(x)$$x$$\Phi(x)$$g$$\Phi(x)$$\alpha,$$\beta$17
such
Proposition 9.1 (K2) There is a family of interval maps
is isomomphic to the lamplighter groupthat the corresponling $G\subset Ant$Z.
$(f, g, \alpha, \beta)$$\lambda^{r}\underline{.)}$$\oplus_{\mathrm{Z}}\mathrm{Z}/\mathrm{Z}_{2})\triangleleft$

ergodically.
acts on 1
In particular for $f,$ $f’,$ above, the group
By dividing $[0, 1]$ into $L+1$ subintcrvals, one can immediately generalize
the above construction. Thus using family of maps, one obtains interaction.
maps $\Phi(x)$ :is an orbit set:
A fiow of
$g,$

$G$

$g’$

$\partial X_{\underline{?}}$$.J\mathrm{Y}_{L+1}arrow d\mathrm{t}_{L+1}^{r}$$\overline{k,}\in X_{L+1}$$\{\Phi(x)^{t}(\overline{k^{4}})\}_{t-- 11}^{\mathrm{x}}-\subset$

Aut$-\mathrm{Y}_{2}$.

The ultra-discrete Lotka Vorterra equation is given by the following families of equations (see [HT]):$v_{l1}^{t+1}-v_{n}^{f}= \max(0, v_{l7+1}’-L)-\max(0, v_{71-1}^{t+1}-L)$.

It is known that they have solitons in solutions.

so
Proposition 9.2 (K2) There is a family of smooth rnapsthat the flow of the corresponding interaction map gives solutions of the
Lotka Vortarra cell automaton.
‘$\{f_{\mathrm{t}.j}\}_{7.j=0_{\backslash\cdots\backslash }I_{\lrcorner}}$

References[CG] A. CARBONE AND M. GROMOV, Mathematical slices of molecular biology, IHES
preprint, (2001).
[HT] R.HIROTA AND D.TAKAHASHI, Discrete and ultradiscrcte systems, in Japan lese,
Kyoritsu shuppan, (2003).

[K1] T.KATO, Operator dynamics in rnolecular biology, IHES preprint (2001).

[K2] T.KATO, Interacting maps, symbolic dynarnics cmd automorphisms in rnolecular biology, Kyoto University preprint (2004).
[UK1,2] K.UENO AND T.KATO EDS., Introduction to molccular biology for mathematicians, in Japanese, (2003,2004).
Tsuyoshi Kato,
Department of Mathem atics, Faculty of Science,
Kyoto University, Kyoto G06-8502, Japan