Wagner's gene network model

Genotype

The model represents individuals as networks of interacting transcriptional regulators. Each individual expresses ${\displaystyle n}$ genes encoding transcription factors. 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 that is represented by a ${\displaystyle N}$ × ${\displaystyle N}$ regulatory matrix ${\displaystyle (R)}$ in the model. The elements in matrix R represent the interaction strength. Positive values within the matrix represent the activation of the target gene, while negative ones represent repression. Matrix elements with value 0 indicate the absence of interactions between two genes.

Phenotype

The phenotype of each individual is modeled as the gene expression pattern at time ${\displaystyle t}$. It is represented by a state vector ${\displaystyle S(t)}$ in this model.

${\displaystyle S(t):=(s_1(t),...,s_N(t))}$

whose element ${\displaystyle s_i(t)}$ denotes the expression state of gene i at time t. In the original Wagner model,

${\displaystyle s_i(t)}$ ∈ ${\displaystyle \{-1,1\}}$

where 1 represents the gene is expressed while -1 implies the gene is not expressed. 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 ${\displaystyle S(0)}$ at time ${\displaystyle t=0}$ is defined as the initial expression state. The interactions among genes change the expression states during the development process. This process is modeled by the following differential equations

${\displaystyle S_l(t+\tau )=\sigma [\sum _{j=1}^N{w}_{ij}{S}_j(t)]=\sigma [h_i](t),}$

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

${\displaystyle \sigma (x)=\{\begin{array}{ll}-1,& (x<0)\\ 1,& (x>0)\\ 0,& (x=0).\end{array}}$

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

${\displaystyle \sigma (x)=\frac{2}{1+e^{-ax}}-1}$

In this way, the expression states will acquire a continuous distribution. The gene expression will reach the final state if it reaches a stable pattern.

Evolution

Evolutionary simulations are performed by reproduction-mutation-selection life cycle. Populations are fixed at size ${\displaystyle N}$ and they will not go extinct. Non-overlapping generations are employed. In a typical evolutionary simulation, a single random viable individual that can produce a stable gene expression pattern is chosen as the founder. Cloned individuals are generated to create a population of ${\displaystyle N}$ identical individuals. According to the asexual or sexual reproductive mode, offspring are produced by randomly choosing (with replacement) parent individual(s) from current generation. Mutations can be acquired with probability μ and survive with probability equal to their fitness. This process is repeated until N individuals are produced that go on to found the following generation.

Fitness

Fitness in this model is the probability that an individual survives to reproduce. In the simplest implementation of the model, developmentally stable genotypes survive (i.e. their fitness is ${\displaystyle 1}$) and developmentally unstable ones do not (i.e. their fitness is ${\displaystyle 0}$).

Mutation

Mutations are modeled as the changes in gene regulation, i.e., the changes of the elements in the regulatory matrix ${\displaystyle R}$.

Reproduction

Both sexual and asexual reproductions are implemented. 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' genomes.

Selection

An organism is considered viable if it reaches a stable gene expression pattern. An organism with oscillated expression pattern is discarded and cannot enter the next generation.