User:Bingjun Zhang/Wagner's gene network model

The Wagner's gene network model was first proposed by Andreas Wagner in 1996 and then used or extended/modified by other groups to study the evolution of gene network, gene expression, robustness, plasticity and epistasis. The model and its variants explicitly modeled the developmental and evolutionary process of genetic regulatory networks.

Assumptions
The model and its variants have a number of simplifying assumptions. Three of them are listing below.
 * 1) The organisms are modeled as gene regulatory networks. The models assume that the gene expression is regulated exclusively on the transcriptional level;
 * 2) The product of one gene could be regulatory factor of itself or other genes. The models assume that one gene can only produce one active transcriptional regulator;
 * 3) The effects of one regulator act independently from other regulator of the same target gene.

Gene Network
The unicellular organisms are modeled as regulatory gene networks with a number $$(N)$$ of interacting genes. The product of each gene can regulate the expression level of itself and/or the other genes through cis-regulatory elements. The interactions among genes constitute a gene network. The gene network is represented by a $$N$$ × $$N$$ regulatory matrix $$(R)$$ in the model. The elements in matrix R represent the interaction strength. Positive values in the matrix represent the active regulation of the target gene, negative ones represent repression.

Gene Expression
Gene expression pattern of a organism at time $$t$$ is represented by a state vector $$S(t)$$

$$S(t)\ :=\ (S_1(t),\ . . .,\ S_N(t))$$ whose elements $$s_i(t)$$ denotes the expression states of gene i at time t. In the original Wagner model,

$$S(t)$$ ∈ $$\{-1,1\} $$

where 1 represents the gene is expressed while -1 is not. The expression pattern can only be ON or OFF. The continuous expression pattern between -1 (or 0) and 1 is also implemented in some other variants.

Development
The development process is modeled as the development of gene expression states. The gene expression pattern $$S(0)$$ at time $$t=0$$ is defined as the initial expression state. Starting from the initial states, the interactions among genes will change the expression states during the development process. This process is modeled by the following difference equations

$$S_l(t+$$τ$$) = $$σ $$[\sum_{j=1}^N w_{ij}S_j(t)]$$

= σ $$[h_i](t)$$

where $$S_l(t+$$τ) represents the expression state of $$G_l$$ at time t+τ. It is determined by a filter function σ$$(x)$$. $$h_i(t)$$ represents the weighted sum of regulatory effects ($$w_{ij}$$) of all genes on gene $$G_l$$ at time t. In the original Wagner model, the filter function is a step function

σ$$(x)=-1$$ if $$x<0;\ 1 $$ if $$x>0;\ 0 $$ if $$x=0$$

In other variants, the filter function is implemented as a sigmoidal function

σ$$(x)=2/(1+e^{-ax})-1$$

In this way, the expression states will acquire a continuous but not discret states. The gene expression will reach the final state if it reach a stable pattern.

Evolutionary Simulation
With this model, a population with multiple organisms can be created and evolved from generation to generation.

Mutation
Mutations are modeled as the changes in gene regulation, i.e., the changes of the elements in the regulatory matrix $$R$$.

Reproduction
Both sexual and asexual reproductions are implemented. The asexual reproduction is implemented as producing the offspring's genome (the gene network) by directly copying the parent's genome. Sexual reproduction is implemented as the recombination of the two parents' genome.

Selection
The organisms are considered to be viable if they reach the stable gene expression pattern but not oscillation. An organism with oscillated expression pattern will be discarded and can not enter the next generation.