Wagner's gene network model

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 networks, 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 gene expression is regulated exclusively at the transcriptional level;
2. The product of a gene can regulate the expression of (be a regulator of) that source gene or other genes. The models assume that a gene can only produce one active transcriptional regulator;
3. The effects of one regulator are independent of effects of other regulators on the same target gene.

Organisms

Gene Network

The figure shows an example of genetic regulatory network and the representation of it in the model, the regulatory matrix R

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

This figure shows an example of how the gene expression pattern is modeled in Wagner model and its variants

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. The interactions among genes change the expression states during the development process. This process is modeled by the following differential 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 discrete states. The gene expression will reach the final state if it reaches a stable pattern.

Evolutionary Simulation

With this model, a population with multiple organisms can be created and evolve 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. 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.

External links

• Andreas Wagner Lab Webpage