DYNAMICS OF RANDOM BOOLEAN NETWORKS

James F. Lynch

Clarkson University

The Lac Operon (Jacob & Monod)

Stuart Kauffman:

Cellular metabolism is controlled by a dynamic

network, where the activity of some genes and

molecules affects the activity of other

constituents of the network.

It is constructed from unreliable parts and is

subjected to mutations, yet it behaves in a

robust and reliable manner.

Is this order and stability the result of

natural selection?

Kauffman: not entirely. There is a statistical

tendency toward order and self-organization.

Natural selection acts on self-organizing

systems, rather than creating them.

Without an innate tendency toward order,

almost all mutations would be fatal.

MODELING CELLULAR METABOLISM AS A BOOLEAN

NETWORK• Gate ≡ Gene or other metabolic element• State ≡ Active/Inactive (1 or 0)• Inputs ≡ Regulators

figate icomputes fi

inputsto

gate i

DYNAMICS

At any time Each gate is in state 0 or 1.Then, each gate reads states of its inputs, say and its state at time becomes .

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STATE SPACE

State of network at time is the vector of states of all gates.

Limit Cycle

state

transition

t

The limit cycle describes the long-term

behavior of the genomic network.

In a multicellular organism, it corresponds

to the cell type after differentiation.

A BOOLEAN NETWORK

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┘

KAUFFMAN’S EXPERIMENTS WITH RANDOM NETWORKS

Number of inputs to each gate Number of gates

For each gate:1. Choose its function from the Boolean

functions of arguments uniformly at random (u.a.r)

2. Choose its inputs u.a.r.3. Choose its starting state u.a.r.

Run the network deterministically

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nk ,k

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k

CLASSIFICATION OF BEHAVIOROrdered:1. Most gates stabilize (stop changing state)

quickly.2. Most gates can be perturbed without

affecting the limit cycle entered.3. Limit cycle is small.

Chaotic:1. Many unstable gates.2. Sensitivity to initial conditions.3. Large limit cycle.

• Ordered behavior is characteristic of genomic and metabolic networks: they quickly settle down into periodic patterns of activity that resist disturbance.

• Chaotic behavior is characteristic of many non-biological complex systems: sensitivity to initial conditions, long transients, and very large limit cycles (strange attractors).

RESULTS OF KAUFFMAN’S SIMULATIONS

• When , the network behaved chaotically.• When , the network exhibited stable

behavior.• Specifically, limit cycle sizes were when

and when .• This is analogous to a phase transition in

dynamical systems.

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2k

3k

n2n 2k

Was the ordered behavior of networks

due to the high proportion of constant Boolean

functions?

A Boolean function is constant if it ignores its

inputs and always outputs the same value, i.e.,

or

Two out of the two argument Boolean

functions are constant, so about 1/8 of the

gates will be assigned constant Boolean

functions.

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0, 21 xxf 1, 21 xxf

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Kauffman also ran simulations of

networks without constant gates: for each

gate, choose its function from the 14 non-

constant Boolean functions of two arguments.

Simulations indicated that behavior was similar

to random Boolean networks that used all

16 Boolean functions of two arguments.

n,2

n,2

Can these results be taken as evidence that:• Biological systems exist at the edge of

chaos?• Self-organization occurs spontaneously in

living systems?• Other researchers have made similar

claims:– Bak (self-organized criticality)– Langton– Packard– Wolfram

MATHEMATICS OF RANDOM BOOLEAN NETWORKS

The behavior of random networks when

or was already known: • networks consist of disjoint cycles of 1-

input gates (identity, negation, and constant gates). They are very stable in all three senses.

• networks behave like random functions on elements. They are very unstable in all three senses. Average state cycle size is

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MORE RECENT RESULTS

• We define a very general class of random Boolean networks that includes the networks as special cases.

• We obtain partial results about the three measures of order on networks in this class.

• Some of our results corroborate Kauffman’s simulations, but some do not.

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DEFINITION OF RANDOM BOOLEAN NETWORK

• Let be an ordering of all finite Boolean functions.

• For each , let be a probability, and letbe the number of arguments of .

• We need some symmetry conditions: whenever and are the same functions, but with re-ordered arguments, or .

• Also and

(mean and variance of the number of arguments is finite).

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CONSTRUCTING A RANDOM BOOLEAN NETWORK WITH

GATESFor each gate ,

1. Assign a Boolean function to , where is the probability that is assigned to .

2. Choose the inputs to uniformly at random.

3. Choose the initial state of uniformly at random.

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ipi

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This kind of random Boolean network includes as special cases:

• Kauffman’s random networks• Networks with classical random graph

topology (Erdős and Rényi) and edge probability

• Networks with power law degree distribution , , or smallworld topology

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DEFINITIONS

• A gate is forced to in steps if its state at any time is , regardless of the initial state of the network.

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• Specifically, let be the Boolean function assigned to .

• is forced to in steps if is the constant function .

• Recursively, is forced to in steps if, letting be its inputs, there is a set

such that for every , is forced to in steps, and for every satisfying for all , .

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For example:

Note that is forced in steps implies stabilizes within steps.

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Forced to 1 in t steps

Forced to 1 in t+1 steps

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• Let be a gate in a Boolean network with gates. Let be the initial state of the network, and . We say that is -weak on input if the state of the network at time is not affected by changing the state of at time .

• Note that is -weak on input implies that perturbing on input will not affect the limit cycle.

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• We will give estimates on the number of gates that are forced in steps and –weak gates, where , based on the distribution .

• In the cases where almost all gates are forced in steps and are –weak, this implies two forms of ordered behavior: most gates will stabilize and most gates can be perturbed without affecting the limit cycle,.

• In some cases we also have estimates on the limit cycle size.

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A PROPERTY OF BOOLEAN FUNCTIONS

Let be a Boolean function with arguments.

Let be a sequence of ’s and

’s.

For we say that argument affects

on input if where when

and .

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EXAMPLES

• Argument 2 affects on input but not on input .

• Argument 1 affects on all inputs.

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Let

is the average number of arguments thataffect on a random input.

Then

is the average number of arguments that affecta random Boolean function on a random input.

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1 2

input on affects

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IS A THRESHOLD FOR FORCED AND WEAK GATES

There is a constant determined by the distribution , such that• If , then with high probability almost all

gates are forced in steps and are -weak.

• If , then with high probability there are at least gates that are not forced in steps and at least gates that are not -weak, where is a constant determined by the distribution .

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APPLICATIONS TO NETWORKS

• When all 16 Boolean functions of two arguments are equally likely, most gates stabilize and are weak.

• When only the 14 non-constant Boolean functions of two arguments are used, instability and sensitivity to initial conditions in the first steps.

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1

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RESULTS ON LIMIT CYCLES IN NETWORKS

• When , with high probability, the limit cycle size is bounded by a constant.

• But when , with high probability, the limit cycle is larger than any polynomial in .

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n,2

COMPARISON TO KAUFFMAN’S SIMULATIONS

• When all 16 2-argument Boolean functions are equally likely, . – Agreement regarding sensitivity to initial conditions and stable

gates.– Disagreement over size of limit cycles: superpolynomial vs.

.

• When only the 14 non-constant functions are used, .– Our results show disorder in the first steps.– But simulations behaved like the case with all 16 Boolean

functions.– This is not necessarily disagreement: the network may settle

down into ordered behavior after steps.

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n

n log

n log

1

SOME PROOF IDEASI. FORCED GATES

• The in-neighborhood of radius of almost all gates is a tree.

AN IN-NEIGHBORHOOD OF RADIUS 3

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• Let be the in-neighborhood of a gate . Assuming is a tree, we extend the notion of “affects” to gates in :

• Gate affects itself if its Boolean function is not a constant.

• Recursively, assume the notion of “affects” has been extended to all gates in that are a distance from . Let be such a gate,

be its Boolean function, and its inputs. For , affects if its corresponding argument affects .

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• Still assuming is a tree, is forced in steps there does not exist a gate in that affects .

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• The recursive definition of “affects” defines a branching process:– For each gate in that affects , its children are

its inputs that affect .

• The expected number of children is .

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• By branching process theory:– If then with probability 1 the process

becomes extinct with high probability, almost all gates are not affected by any gate in theirin-neighborhood they are forced in steps.

– If then with probability 1 the process will not become extinct with high probability, approximately gates are affected by some gate in their in-neighborhood they are not forced in steps.

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II. WEAK GATES

• The out-neighborhood of radius of almost all gates is a tree.

• This means that the effect of most perturbations is approximated by a branching process.

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• Assume that the out-neighborhood of a gate is a tree.

• The probability that perturbing a gate in this out-neighborhood affects of its children is

• the expected number of children affected

by the gate is .

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!

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• Again by branching process theory:– If then with probability 1 the process

becomes extinct with high probability, for almost all gates, the effect of the perturbation disappears.

– If then with probability the process will not become extinct with high probability, for approximately gates, the effect of a perturbation will persist for at least steps.

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OPEN PROBLEMS

• Do many perturbations of the initial state cause permanent changes in the state when ?

• What is the limit cycle size when ?• Networks with external inputs and outputs:

– What kinds of functions can be computed?– Is a region where complex functions are

computed?

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1

1

• More general kinds of networks:– Real-valued states of gates– Asynchronous dynamics– Continuous dynamics– Probabilistic dynamics

SUMMARY

• Simulations of complex systems may not be reliable. If possible, they should be verified with analytic results.

• Dynamical systems approach needs to be applied to more realistic, detailed models.– In particular, the notion of “complexity at the edge

of chaos” should be tested against specific systems, such as models of self-assembling membranes or other cellular organelles.

• More generally, we’ve presented a form of Individual-Based Model, where the individuals are gates, the population is described by the functions of the gates and their connections, and we are interested in statistical properties of its dynamics.

• Individual-Based Models are now being used to model populations at all scales in biology:– Bray & Firth: StochSim stochastic simulator of

molecular reactions– Romey: fish & whirligigs

• Can a theory of the dynamics of Individual-Based Models be developed?