^{1}

^{1}

^{1}

^{1}

^{2}

^{*}

Conceived and designed the experiments: NK AEM TK UA. Performed the experiments: NK AEM TK. Analyzed the data: NK AEM TK UA. Wrote the paper: NK AEM TK UA.

The authors have declared that no competing interests exist.

Biological systems often display modularity, in the sense that they can be decomposed into nearly independent subsystems. Recent studies have suggested that modular structure can spontaneously emerge if goals (environments) change over time, such that each new goal shares the same set of sub-problems with previous goals. Such modularly varying goals can also dramatically speed up evolution, relative to evolution under a constant goal. These studies were based on simulations of model systems, such as logic circuits and RNA structure, which are generally not easy to treat analytically. We present, here, a simple model for evolution under modularly varying goals that can be solved analytically. This model helps to understand some of the fundamental mechanisms that lead to rapid emergence of modular structure under modularly varying goals. In particular, the model suggests a mechanism for the dramatic speedup in evolution observed under such temporally varying goals.

Biological systems often display modularity, in the sense that they can be decomposed into nearly independent subsystems. The evolutionary origin of modularity has recently been the focus of renewed attention. A series of studies suggested that modularity can spontaneously emerge in environments that vary over time in a modular fashion—goals composed of the same set of subgoals but each time in a different combination. In addition to spontaneous generation of modularity, evolution was found to be dramatically accelerated under such varying environments. The time to achieve a given goal was much shorter under varying environments in comparison to constant conditions. These studies were based on computer simulations of simple model systems such as logic circuits and RNA secondary structure. Here, we take this a step forward. We present a simple mathematical model that can be solved analytically and suggests mechanisms that lead to the rapid emergence of modular structure.

Biological systems often display modularity, defined as the seperability of the
design into units that perform independently, at least to a first approximation

The evolution of modularity has been a puzzle because computer simulations of
evolution are well-known to lead to non-modular solutions. This tendency of
simulations to evolve non-modular structures is familiar in fields such as evolution
of neural networks, evolution of hardware and evolution of software. In almost all
cases, the evolved systems cannot be decomposed into sub-systems, and are difficult
to understand intuitively

Several suggestions have been made to address the origin of modularity in biological
evolution

Examples of data from a series of studies _{1} = (x XOR
y) AND (w XOR z). The circuit is composed of 10 NAND gates. Evolution under
a constant goal typically yields compact non-modular circuits. (B) Circuits
evolved under MVG evolution, varying every 20 generations between goal
_{1} and goal
_{2} = (x XOR y)
OR (w XOR z). Note that these two goals share the same sub-goals, namely two
XOR functions. Connections that are rewired when the goal switches are
marked in red. Evolution under MVG typically yields modular circuits that
are less compact, composed in this case of 11 gates. The circuits are
composed of three modules: two XOR modules and a third module that
implements an AND/OR function, depending on the goal.

In addition to promoting modularity, MVG was also found to dramatically speed
evolution relative to evolution under a constant goal

(A) A schematic view of fitness as a function of generations in evolution
under MVG and fixed (constant) goal (FG). Evolution time
(_{MVG}_{FG}_{FG}/T_{MVG}_{FG}_{FG})
^{α}

To summarize the main findings of

A constant goal (that does not change over time) leads to non-modular structures.

Modularly varying goals lead to modular structures.

Evolution converges under MVG much faster than under a constant goal.

The harder the goals, the faster the speedup observed in MVG relative to constant goal evolution.

Random (non-modular) goals that vary over time usually lead to evolutionary confusion without generating modular structure, and rarely lead to speedup.

Since these findings were based on simulations, it is of interest to try to find a model that can be solved analytically so that the reasons for the emergence of modular structure, and for the speedup of evolution, can be more fully understood. Here we present such a simple, exactly solvable model. The model allows one to understand some of the mechanisms that lead to modularity and speedup in evolution.

The guiding principle in building the model was to find the simplest system that
shows the salient features described in the

The matrix

An evolutionary goal in the present study is that an input vector

To evaluate the fitness of the system, we follow experimental studies in
bacteria, that suggest that biological circuits can be assigned benefit and cost

We begin with the cost of the system, related to the magnitude of the elements of
_{ij}_{ij}

In addition to the cost, each structure has a benefit. The benefit
_{o}

The first term on the right hand side represents the cost of the elements of

In realistic situations, the parameter

Now that we have defined the fitness function, we turn to the definition of modularity in structures and in goals.

A modular structure, which corresponds to a modular matrix

The _{m}_{m}_{m}_{m}_{m}

In addition to the modularity of the structure _{o}

Here the first component of each output vector is a linear function of the first
component of the corresponding input vectors, namely the identity function. The
next two components of each output vector are equal to a linear 2×2
matrix,
_{1} = u_{1}_{2} = u_{2}_{o} can be
decomposed into independent groups of components, using the same linear
functions. Hence, the goal _{o} is modular. Note that
most goals (most input-output vector sets with

To quantify the modularity of a structure _{m}

In the following sections we analyze the dynamics and convergence of evolution under
both fixed goal conditions and under MVG conditions. For clarity we first present a
two–dimensional system
(

We begin with two-dimensional system
(

Consider the goal _{1}_{1}

Let us find the most fit structure _{1}_{ij}

Solving these equations, we find that the highest fitness structure is

Note that indeed,
_{m}^{*}

It is also helpful to graphically display this solution. _{11}_{12}_{11}+a_{12}_{11}_{12}^{*}_{11},
a_{12}

(A) Matrix elements are portrayed in a two dimensional space defined
by _{11}_{12}_{1}_{12}_{11}_{1}. Black dots display the dynamics
at 100/r time unit resolution, where _{2}

A non-modular solution is the general solution for this type of goal (proof
in section _{2}_{1}, the highest-fitness structure
for G_{2} is non-modular, (

We now turn to discuss the dynamics of the evolutionary process. We ask how
long it takes to reach the maximum-fitness structure starting from a random
initial structure. For this purpose, one needs to define the dynamics of
evolutionary change and selection. For simplicity, we consider a
Hill-climbing picture, in which the rate of change of the structure
_{ij}

The Hill-climbing dynamical model is simple enough to analytically solve for
the dynamics of the matrix elements _{ij}

These are linear ordinary differential equations, and hence the solution for
_{ij}_{n}
_{n}_{n}_{ij}

The convergence times are thus governed by the eignevalues
_{n}

For example, for the goal _{1}_{1} = _{2} = _{3} = _{4} = _{3} and
_{4} correspond to rapid evolution to
the line shown in _{1} = _{2} = _{1}
t_{FG}∼1/λ_{1}∼1/ε_{2}_{1} = _{2} =

We next consider the case where the environment changes over time, switching
between the two modular goals mentioned above. For example, the structure
_{1}_{1}_{1}_{2}_{2}_{2}_{1}_{2}_{1}_{2}

What is the structure that evolves under MVG? We use the dynamical equations
(Eq. 4) to describe the MVG process which switches between the goals.

Here Eqs. 6a and 6b are valid for times when the goals are
_{1}_{2}_{1}_{1}_{2}_{2}

Goals are switched between
_{1} = [_{2} = [

To analyze this scenario, consider the limiting case where switches between
the two goals occur very rapidly. In this case, one can average the fitness
over time, and ask which structure maximizes the average fitness. If the
environment spends, say, half of the time with goal
_{1}_{2}

One can then solve the equations for the elements
_{ij}

Intuitively, supplying two modular goals provides ‘extra
information’ that helps evolution find the unique structure that
satisfies both goals – even though the different goals do not
appear at the same time. If one stops varying the goals and presents a
constant goal _{1}_{2}

We have seen that MVG leads to a modular structure. Let us now analyze the
time that it takes the evolutionary process to approach this modular
solution, starting from a random initial condition. In contrast to the small
eigenvalues (long convergence time) found under a constant goal, a different
situation is found under MVG. Here, evolution converges rapidly to the
modular solution, with convergence time of order one
_{MVG}∼1

To understand why dynamics are rapid, consider the view depicted in _{FG}
∼1/ε

It is also helpful to visually examine the fitness landscapes that govern the
dynamics of MVG. One can get a feeling for the shape of the landscape by
looking at the fitness function averaged over both goals. The rapid
convergence to a modular solution is due to the formation of a steep peak in
the ‘effective’ combined fitness landscape, as opposed
to a flat ridge in the case of evolution under a constant goal (

Goals _{1}_{2}_{21}, a_{22}_{1}_{2}_{1}_{2}_{1}_{2}

The two dimensional case we have discussed is relatively easy to visualize.
Let us now consider higher dimensions. We will consider a three-dimensional
problem (

Let us begin with the goal _{1}

Note that _{1}_{1}_{1}

The dynamical equations have a small eigenvalue
_{FG}∼1/ε

Presented is the three dimensional space defined by
_{11}_{12}_{13}_{1} = _{1} and
_{2} = _{2}).
(A) A Constant goal
_{1}_{11} = (1,−1,−1.4),
_{11} = (1,−2.4,0.4)];
[_{12} = (0.5,1.2,−1.9),
_{12} = (0.5,−0.7,3.1)
] }. (B) Modularly varying goals.
_{1} as above, and
_{2}_{11} = (1,1.7,−0.7),
_{11} = (1,1,2.4)
]; [
_{12} = (−0.7,−2.3,−1.1),
_{12} = (−0.7,−3.4,−1.2)
] }. Switching rate is
_{1}_{2}_{1}_{2} = { [
(1.1,1,1), (1.1,1,1) ]; [ (0.6,−1,1),
(0.6,−1,1) ] }.

In contrast, if MVG is applied, switching between
_{1}_{2}

Modularity increases rapidly as shown in

Modularity of the system measured by normalized community structure
_{m}

Up to now, the varying goals shared the same modular solution. Let us
consider a more general case where the varying goals
_{1}_{2}

As an example, which represents the typical case, let
_{1}_{2}

We find that evolution under varying goals in such cases rapidly leads to a
structure that is modular. Once the modular structure was established, the
system moves between the two similar modular matrices every time the goal
switches (

(A) Speedup as a function of goal switching times _{1}_{1}_{2} :
G_{1}_{2}_{FG}
/ T_{MVG}_{FG}_{FG}

What is the effect of switching time (rate at which goals are switched) on
the speedup? We find that speedup is high over a wide range of switching
times. Speedup occurs provided that the switching times

In the case of nearly-modular varying goals, speedup occurs provided that
epoch times

We briefly consider also a higher dimensional example with
_{1}_{2}

MVG evolution with these two goals converges to a block-modular structure

At this point, it is interesting to note that, in all dimensions, the block
structure of the evolved matrix relates to the correlations within the goal
input and output vectors. In fact, the block structure of _{11},
v_{12}, v_{21}, v_{22}_{11},
u_{12}, u_{21}, u_{22}

What happens to modularity under a constant goal if one begins with a modular
solution as an initial condition? We find that modularity decays over time
(

So far, we have considered modularly varying goals - that is goals that have
a special feature: their components can be decomposed into modules, with the
same (or nearly the same) modules for all goals. Thus, there exists a
modular matrix

Pairs of randomly chosen goals (with

It is easy to understand this using a geometrical picture. One can represent
the set of solutions for each goal as a line (or hyper-plane) in the space
of matrix elements. The solution lines of two random goals in the high
dimension space have very low probability to cross or even to come close to
each other. Switching between goals generally leads to a motion around the
point where the lines come closest, which is generally a rather poor
solution for each of the goals (

Such confusion is avoided in the case of MVG, because goals share the same
(or nearly the same) modular structure. Such a set of modular goals is
special: it ensures that the corresponding lines intersect (or nearly
intersect), and in particular that they intersect on one of the axes. One
can prove (see section

There are special cases in which the goals are non-modular but still afford a
speedup in evolution. This happens when the goal vectors happen to be
linearly dependent such that a non-modular structure

One can define the

As pointed out above, the convergence time in a fixed goal (with dynamics
mostly _{FG}∼1/ε_{MVG}∼1/λ

Thus, the ‘harder’ the fixed goal problem is (that is,
the smaller _{FG}_{FG})^{α}

Here we calculate the optimal solution in a problem in which the goal is fixed (FG), and in a problem with modularly varying goals (MVG). We show that the fitness of the optimal solution in a FG problem is higher than the fitness of the solution in a MVG problem.

We begin by considering the fitness function of Eq. 2 written in matrix form.

Here

The equation of motion for _{N} is the
^{T}^{T}

The optimum of

Taking the limit ^{+}^{*} = MVV^{+}

The solution in an MVG problem with

Here we assume that equal amounts of time are spent in each goal. If this is
not the case then the average over goals should be replaced by a weighted
mean. Eq. (A6) can be further simplified by noting that

Here

With this, the equation of motion reads

We assume that

The fitness,

Here we used the inequality _{m}<F^{*}

First we show that goals with

We begin by writing the solution of Eq. (A2):

We will show now that

Here _{k}^{T}
^{T}

Using the rule

Geometrically, this means that the dynamics in the

For completeness we write the solution (B1) in terms of the eigensystem of
the coefficient matrix:

Note that this solution holds for MVG problems. At the beginning of each
epoch (after a goal switch) we update the initial conditions (equal to the
value of the matrix

Now we show that in an MVG problem with

We approach this problem by taking the limit of vanishing small switching
time. In this case the MVG problem is equivalent to the average problem with
the equation of motion Eq. (A8). Thus the eigensystem in this case is
determined by the characteristic polynomial of the average problem:

In the generic case

Geometrically, this means that unlike the dynamics in a FG problem, the
dynamics in an MVG problem in

For completeness we write the solution for the equation of motion (A8)

We studied a model for evolution under temporally varying goals that can be exactly
solved. This model captures some of the features previously observed with
simulations of more complex systems

The speedup of evolution under MVG is a phenomenon that was previously found using
simulations, but lacked an analytical understanding. The present model offers an
analytical explanation for the speedup observed under MVG. The speedup in the model
is related to small eigenvalues that correspond to motion along fitness plateaus
when the goal is constant in time. These eigenvalues become large when the goal
changes over time, because in MVG, the plateaus of one goal become a high-slope
fitness region for the other goal. Switching between goals guides evolution along a
‘ramp’ that leads to the modular solution. This analytical
solution of the dynamics agrees with the qualitative analysis based on sampling of
the fitness landscape during the evolutionary simulations of complex models

One limitation in comparing the present model to more complex simulations is that the present model lacks a complex fitness landscape with many plateaus and local maxima. Such plateaus and local fitness maxima make constant-goal evolution even more difficult, and are expected to further augment the speed of MVG relative to constant goal conditions. A second limitation of the present linear model is that it can solve different MVG goals when presented simultaneously - a feature not possible for nonlinear systems. This linearity of the model, however, provides a clue to how MVG evolution works: whereas each goal supplies only partial information, all goals together specify the unique modular solution. Under MVG evolution, the system effectively remembers previous goals, supplying the information needed to guide evolution to the modular solution, even though at each time point the current goal provides insufficient information. This memory effect is likely to occur in the nonlinear systems as well.

The series of studies on MVG, including the present theory, predict that organisms or
molecules whose environment does not change over time should gradually lose their
modular structure and approach a non-modular (but more optimal) structure. This
suggestion was supported by a study that showed that bacteria that live in
relatively constant niches such as obligate parasites that live inside cells, seem
to have a less modular metabolic network than organisms in varying environments such
as the soil

In summary, the present model provides an analytical explanation for the evolution of
modular structures and for the speedup of evolution under MVG, previously found by
means of simulations. In the present view, the modularity of evolved structures is
an internal representation of the modularity found in the world

A Simple Model for Rapid Evolution of Modularity

(0.42 MB PDF)

We thank Elad Noor, Merav Parter, Yuval Hart and Guy Shinar for comments and discussions.