Gene regulatory network

Global feature

Gene regulatory networks are generally thought to be made up of a few highly connected nodes (hubs) and many poorly connected nodes nested within a hierarchical regulatory regime. Thus gene regulatory networks approximate a hierarchical scale free network topology.^[8] This is consistent with the view that most genes have limited pleiotropy and operate within regulatory modules.^[9] This structure is thought to evolve due to the preferential attachment of duplicated genes to more highly connected genes.^[8] Recent work has also shown that natural selection tends to favor networks with sparse connectivity.^[10]

There are primarily two ways that networks can evolve, both of which can occur simultaneously. The first is that network topology can be changed by the addition or subtraction of nodes (genes) or parts of the network (modules) may be expressed in different contexts. The Drosophila Hippo signaling pathway provides a good example. The Hippo signaling pathway controls both mitotic growth and post-mitotic cellular differentiation.^[11] Recently it was found that the network the Hippo signaling pathway operates in differs between these two functions which in turn changes the behavior of the Hippo signaling pathway. This suggests that the Hippo signaling pathway operates as a conserved regulatory module that can be used for multiple functions depending on context.^[11] Thus, changing network topology can allow a conserved module to serve multiple functions and alter the final output of the network. The second way networks can evolve is by changing the strength of interactions between nodes, such as how strongly a transcription factor may bind to a cis-regulatory element. Such variation in strength of network edges has been shown to underlie between species variation in vulva cell fate patterning of Caenorhabditis worms.^[12]

Local feature

Another widely cited characteristic of gene regulatory network is their abundance of certain repetitive sub-networks known as network motifs. Network motifs can be regarded as repetitive topological patterns when dividing a big network into small blocks. Previous analysis found several types of motifs that appeared more often in gene regulatory networks than in randomly generated networks.^[13][14][15] As an example, one such motif is called feed-forward loops, which consist of three nodes. This motif is the most abundant among all possible motifs made up of three nodes, as is shown in the gene regulatory networks of fly, nematode, and human.^[15]

The enriched motifs have been proposed to follow convergent evolution, suggesting they are "optimal designs" for certain regulatory purposes.^[16] For example, modeling shows that feed-forward loops are able to coordinate the change in node A (in terms of concentration and activity) and the expression dynamics of node C, creating different input-output behaviors.^[17][18] The galactose utilization system of E. coli contains a feed-forward loop which accelerates the activation of galactose utilization operon galETK, potentially facilitating the metabolic transition to galactose when glucose is depleted.^[19] The feed-forward loop in the arabinose utilization systems of E.coli delays the activation of arabinose catabolism operon and transporters, potentially avoiding unnecessary metabolic transition due to temporary fluctuations in upstream signaling pathways.^[20] Similarly in the Wnt signaling pathway of Xenopus, the feed-forward loop acts as a fold-change detector that responses to the fold change, rather than the absolute change, in the level of β-catenin, potentially increasing the resistance to fluctuations in β-catenin levels.^[21] Following the convergent evolution hypothesis, the enrichment of feed-forward loops would be an adaptation for fast response and noise resistance. A recent research found that yeast grown in an environment of constant glucose developed mutations in glucose signaling pathways and growth regulation pathway, suggesting regulatory components responding to environmental changes are dispensable under constant environment.^[22]

On the other hand, some researchers hypothesize that the enrichment of network motifs is non-adaptive.^[23] In other words, gene regulatory networks can evolve to a similar structure without the specific selection on the proposed input-output behavior. Support for this hypothesis often comes from computational simulations. For example, fluctuations in the abundance of feed-forward loops in a model that simulates the evolution of gene regulatory networks by randomly rewiring nodes may suggest that the enrichment of feed-forward loops is a side-effect of evolution.^[24] In another model of gene regulator networks evolution, the ratio of the frequencies of gene duplication and gene deletion show great influence on network topology: certain ratios lead to the enrichment of feed-forward loops and create networks that show features of hierarchical scale free networks. De novo evolution of coherent type 1 feed-forward loops has been demonstrated computationally in response to selection for their hypothesized function of filtering out a short spurious signal, supporting adaptive evolution, but for non-idealized noise, a dynamics-based system of feed-forward regulation with different topology was instead favored.^[25]

Coupled ordinary differential equations

It is common to model such a network with a set of coupled ordinary differential equations (ODEs) or SDEs, describing the reaction kinetics of the constituent parts. Suppose that our regulatory network has ${\displaystyle N}$ nodes, and let ${\displaystyle S_{1}(t),S_{2}(t),\ldots ,S_{N}(t)}$ represent the concentrations of the ${\displaystyle N}$ corresponding substances at time ${\displaystyle t}$. Then the temporal evolution of the system can be described approximately by

${\displaystyle {\frac {dS_{j}}{dt}}=f_{j}\left(S_{1},S_{2},\ldots ,S_{N}\right)}$

where the functions ${\displaystyle f_{j}}$ express the dependence of ${\displaystyle S_{j}}$ on the concentrations of other substances present in the cell. The functions ${\displaystyle f_{j}}$ are ultimately derived from basic principles of chemical kinetics or simple expressions derived from these e.g. Michaelis–Menten enzymatic kinetics. Hence, the functional forms of the ${\displaystyle f_{j}}$ are usually chosen as low-order polynomials or Hill functions that serve as an ansatz for the real molecular dynamics. Such models are then studied using the mathematics of nonlinear dynamics. System-specific information, like reaction rate constants and sensitivities, are encoded as constant parameters.^[30]

By solving for the fixed point of the system:

${\displaystyle {\frac {dS_{j}}{dt}}=0}$

for all ${\displaystyle j}$, one obtains (possibly several) concentration profiles of proteins and mRNAs that are theoretically sustainable (though not necessarily stable). Steady states of kinetic equations thus correspond to potential cell types, and oscillatory solutions to the above equation to naturally cyclic cell types. Mathematical stability of these attractors can usually be characterized by the sign of higher derivatives at critical points, and then correspond to biochemical stability of the concentration profile. Critical points and bifurcations in the equations correspond to critical cell states in which small state or parameter perturbations could switch the system between one of several stable differentiation fates. Trajectories correspond to the unfolding of biological pathways and transients of the equations to short-term biological events. For a more mathematical discussion, see the articles on nonlinearity, dynamical systems, bifurcation theory, and chaos theory.

Boolean network

The following example illustrates how a Boolean network can model a GRN together with its gene products (the outputs) and the substances from the environment that affect it (the inputs). Stuart Kauffman was amongst the first biologists to use the metaphor of Boolean networks to model genetic regulatory networks.^[31][32]

1. Each gene, each input, and each output is represented by a node in a directed graph in which there is an arrow from one node to another if and only if there is a causal link between the two nodes.
2. Each node in the graph can be in one of two states: on or off.
3. For a gene, "on" corresponds to the gene being expressed; for inputs and outputs, "on" corresponds to the substance being present.
4. Time is viewed as proceeding in discrete steps. At each step, the new state of a node is a Boolean function of the prior states of the nodes with arrows pointing towards it.

The validity of the model can be tested by comparing simulation results with time series observations. A partial validation of a Boolean network model can also come from testing the predicted existence of a yet unknown regulatory connection between two particular transcription factors that each are nodes of the model.^[33]

Continuous networks

Continuous network models of GRNs are an extension of the Boolean networks described above. Nodes still represent genes and connections between them regulatory influences on gene expression. Genes in biological systems display a continuous range of activity levels and it has been argued that using a continuous representation captures several properties of gene regulatory networks not present in the Boolean model.^[34] Formally most of these approaches are similar to an artificial neural network, as inputs to a node are summed up and the result serves as input to a sigmoid function, e.g.,^[35] but proteins do often control gene expression in a synergistic, i.e. non-linear, way.^[36] However, there is now a continuous network model^[37] that allows grouping of inputs to a node thus realizing another level of regulation. This model is formally closer to a higher order recurrent neural network. The same model has also been used to mimic the evolution of cellular differentiation^[38] and even multicellular morphogenesis.^[39]

Stochastic gene networks

Experimental results^{[40] [41]} have demonstrated that gene expression is a stochastic process. Thus, many authors are now using the stochastic formalism, after the work by Arkin et al.^[42] Works on single gene expression^[43] and small synthetic genetic networks,^[44][45] such as the genetic toggle switch of Tim Gardner and Jim Collins, provided additional experimental data on the phenotypic variability and the stochastic nature of gene expression. The first versions of stochastic models of gene expression involved only instantaneous reactions and were driven by the Gillespie algorithm.^[46]

Since some processes, such as gene transcription, involve many reactions and could not be correctly modeled as an instantaneous reaction in a single step, it was proposed to model these reactions as single step multiple delayed reactions in order to account for the time it takes for the entire process to be complete.^[47]

From here, a set of reactions were proposed^[48] that allow generating GRNs. These are then simulated using a modified version of the Gillespie algorithm, that can simulate multiple time delayed reactions (chemical reactions where each of the products is provided a time delay that determines when will it be released in the system as a "finished product").

For example, basic transcription of a gene can be represented by the following single-step reaction (RNAP is the RNA polymerase, RBS is the RNA ribosome binding site, and Pro _i is the promoter region of gene i):

${\displaystyle {\text{RNAP}}+{\text{Pro}}_{i}{\overset {k_{i,bas}}{\longrightarrow }}{\text{Pro}}_{i}(\tau _{i}^{1})+{\text{RBS}}_{i}(\tau _{i}^{1})+{\text{RNAP}}(\tau _{i}^{2})}$

Furthermore, there seems to be a trade-off between the noise in gene expression, the speed with which genes can switch, and the metabolic cost associated their functioning. More specifically, for any given level of metabolic cost, there is an optimal trade-off between noise and processing speed and increasing the metabolic cost leads to better speed-noise trade-offs.^[49][50][51]

A recent work proposed a simulator (SGNSim, Stochastic Gene Networks Simulator),^[52] that can model GRNs where transcription and translation are modeled as multiple time delayed events and its dynamics is driven by a stochastic simulation algorithm (SSA) able to deal with multiple time delayed events. The time delays can be drawn from several distributions and the reaction rates from complex functions or from physical parameters. SGNSim can generate ensembles of GRNs within a set of user-defined parameters, such as topology. It can also be used to model specific GRNs and systems of chemical reactions. Genetic perturbations such as gene deletions, gene over-expression, insertions, frame shift mutations can also be modeled as well.

The GRN is created from a graph with the desired topology, imposing in-degree and out-degree distributions. Gene promoter activities are affected by other genes expression products that act as inputs, in the form of monomers or combined into multimers and set as direct or indirect. Next, each direct input is assigned to an operator site and different transcription factors can be allowed, or not, to compete for the same operator site, while indirect inputs are given a target. Finally, a function is assigned to each gene, defining the gene's response to a combination of transcription factors (promoter state). The transfer functions (that is, how genes respond to a combination of inputs) can be assigned to each combination of promoter states as desired.

In other recent work, multiscale models of gene regulatory networks have been developed that focus on synthetic biology applications. Simulations have been used that model all biomolecular interactions in transcription, translation, regulation, and induction of gene regulatory networks, guiding the design of synthetic systems.^[53]

Multiple sclerosis

There are three classes of multiple sclerosis: relapsing-remitting (RRMS), primary progressive (PPMS) and secondary progressive (SPMS). Gene regulatory network (GRN) plays a vital role to understand the disease mechanism across these three different multiple sclerosis classes.^[57]