Editing Gene regulatory network (section)

== Modelling ==

=== Coupled ordinary differential equations ===
{{for|an example of modelling of the cell cycle with ODEs|cellular model}}

It is common to model such a network with a set of coupled [[ordinary differential equation]]s (ODEs) or [[Stochastic differential equation|SDE]]s, describing the reaction kinetics of the constituent parts. Suppose that our regulatory network has <math>N</math> nodes, and let <math>S_1(t),S_2(t), \ldots, S_N(t)</math> represent the concentrations of the <math>N</math> corresponding substances at time <math>t</math>. Then the temporal evolution of the system can be described approximately by

: <math> \frac{dS_j}{dt} = f_j \left (S_1,S_2, \ldots, S_N \right) </math>

where the functions <math> f_j </math> express the dependence of <math>S_j</math> on the concentrations of other substances present in the cell. The functions <math>f_j</math> are ultimately derived from basic [[rate equation|principles of chemical kinetics]] or simple expressions derived from these e.g. [[Michaelis–Menten]] enzymatic kinetics. Hence, the functional forms of the <math>f_j</math> are usually chosen as low-order [[polynomials]] or [[Hill equation (biochemistry)|Hill function]]s that serve as an [[ansatz]] for the real molecular dynamics. Such models are then studied using the mathematics of [[dynamical system|nonlinear dynamics]]. System-specific information, like [[reaction rate]] constants and sensitivities, are encoded as constant parameters.<ref>{{cite journal | vauthors = Chu D, Zabet NR, Mitavskiy B | title = Models of transcription factor binding: sensitivity of activation functions to model assumptions | journal = Journal of Theoretical Biology | volume = 257 | issue = 3 | pages = 419–429 | date = April 2009 | pmid = 19121637 | doi = 10.1016/j.jtbi.2008.11.026 | bibcode = 2009JThBi.257..419C | s2cid = 12809260 | url = https://kar.kent.ac.uk/24077/1/myzabetpaper.pdf }}</ref>

By solving for the [[Fixed point (mathematics)|fixed point]] of the system:

: <math> \frac{dS_j}{dt} = 0 </math>

for all <math>j</math>, one obtains (possibly several) concentration profiles of proteins and mRNAs that are theoretically sustainable (though not necessarily [[stability (mathematics)|stable]]). [[Steady state]]s of kinetic equations thus correspond to potential cell types, and [[Oscillation|oscillatory]] solutions to the above equation to naturally cyclic cell types. Mathematical stability of these [[attractor]]s can usually be characterized by the sign of higher derivatives at critical points, and then correspond to [[Steady state (biochemistry)|biochemical stability]] of the concentration profile. [[Critical point (mathematics)|Critical point]]s and [[Bifurcation theory|bifurcation]]s 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.<ref name="kauffmanRBN">{{cite book | title=The Origins of Order| vauthors = Kauffman SA | year=1993 | publisher = Oxford University Press | isbn=978-0-19-505811-6}}</ref><ref>{{cite journal | vauthors = Kauffman SA | title = Metabolic stability and epigenesis in randomly constructed genetic nets | journal = Journal of Theoretical Biology | volume = 22 | issue = 3 | pages = 437–467 | date = March 1969 | pmid = 5803332 | doi = 10.1016/0022-5193(69)90015-0 | bibcode = 1969JThBi..22..437K }}</ref>

# 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.
# Each node in the graph can be in one of two states: on or off.
# For a gene, "on" corresponds to the gene being expressed; for inputs and outputs, "on" corresponds to the substance being present.
# 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.<ref name="pmid25398016">{{cite journal | vauthors = Lovrics A, Gao Y, Juhász B, Bock I, Byrne HM, Dinnyés A, Kovács KA | title = Boolean modelling reveals new regulatory connections between transcription factors orchestrating the development of the ventral spinal cord | journal = PLOS ONE | volume = 9 | issue = 11 | pages = e111430 | date = November 2014 | pmid = 25398016 | pmc = 4232242 | doi = 10.1371/journal.pone.0111430 | doi-access = free | bibcode = 2014PLoSO...9k1430L }}</ref>

=== 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.<ref>{{cite journal | vauthors = Vohradsky J | title = Neural model of the genetic network | journal = The Journal of Biological Chemistry | volume = 276 | issue = 39 | pages = 36168–36173 | date = September 2001 | pmid = 11395518 | doi = 10.1074/jbc.M104391200 | doi-access = free }}

</ref> 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.,<ref>{{cite journal | vauthors = Geard N, Wiles J | title = A gene network model for developing cell lineages | journal = Artificial Life | volume = 11 | issue = 3 | pages = 249–267 | year = 2005 | pmid = 16053570 | doi = 10.1162/1064546054407202 | s2cid = 8664677 | citeseerx = 10.1.1.1.4742 }}</ref> but proteins do often control gene expression in a synergistic, i.e. non-linear, way.<ref>{{cite web |vauthors=Schilstra MJ, Bolouri H |title=Modelling the Regulation of Gene Expression in Genetic Regulatory Networks |date=2 January 2002 |publisher=Biocomputation group, University of Hertfordshire |url=http://strc.herts.ac.uk/bio/maria/NetBuilder/Theory/NetBuilderModelling.htm |url-status=dead |archive-url=https://web.archive.org/web/20071013022705/http://strc.herts.ac.uk/bio/maria/NetBuilder/Theory/NetBuilderModelling.htm |archive-date=13 October 2007 |df=dmy }}</ref> However, there is now a continuous network model<ref>{{cite conference |vauthors=Knabe JF, Nehaniv CL, Schilstra MJ, Quick T |title=Evolving Biological Clocks using Genetic Regulatory Networks |book-title=Proceedings of the Artificial Life X Conference (Alife 10) |pages=15–21 |publisher=MIT Press |year=2006 |citeseerx = 10.1.1.72.5016 }}</ref> 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]]<ref>{{cite conference |vauthors=Knabe JF, Nehaniv CL, Schilstra MJ |title=Evolutionary Robustness of Differentiation in Genetic Regulatory Networks |book-title=Proceedings of the 7th German Workshop on Artificial Life 2006 (GWAL-7) |pages=75–84 |publisher=[[Akademische Verlagsgesellschaft AKA]] |year=2006 |location=Berlin |citeseerx = 10.1.1.71.8768 }}</ref> and even multicellular [[morphogenesis]].<ref>{{cite conference |vauthors=Knabe JF, Schilstra MJ, Nehaniv CL |title=Evolution and Morphogenesis of Differentiated Multicellular Organisms: Autonomously Generated Diffusion Gradients for Positional Information |book-title=Artificial Life XI: Proceedings of the Eleventh International Conference on the Simulation and Synthesis of Living Systems |publisher=MIT Press |year=2008 |url=http://panmental.de/papers/FlagPottsGRNALife11.pdf }}</ref>

=== Stochastic gene networks ===

Experimental results<ref>{{cite journal | vauthors = Elowitz MB, Levine AJ, Siggia ED, Swain PS | title = Stochastic gene expression in a single cell | journal = Science | volume = 297 | issue = 5584 | pages = 1183–1186 | date = August 2002 | pmid = 12183631 | doi = 10.1126/science.1070919 | s2cid = 10845628 | bibcode = 2002Sci...297.1183E | url = https://authors.library.caltech.edu/records/wsymf-b6c81/files/ElowitzSOM.pdf?download=1 }}</ref>
<ref>{{cite journal | vauthors = Blake WJ, KAErn M, Cantor CR, Collins JJ | title = Noise in eukaryotic gene expression | journal = Nature | volume = 422 | issue = 6932 | pages = 633–637 | date = April 2003 | pmid = 12687005 | doi = 10.1038/nature01546 | s2cid = 4347106 | bibcode = 2003Natur.422..633B }}</ref> 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.<ref>{{cite journal | vauthors = Arkin A, Ross J, McAdams HH | title = Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells | journal = Genetics | volume = 149 | issue = 4 | pages = 1633–1648 | date = August 1998 | pmid = 9691025 | pmc = 1460268 | doi = 10.1093/genetics/149.4.1633 }}</ref> Works on single gene expression<ref>{{cite journal | vauthors = Raser JM, O'Shea EK | title = Noise in gene expression: origins, consequences, and control | journal = Science | volume = 309 | issue = 5743 | pages = 2010–2013 | date = September 2005 | pmid = 16179466 | pmc = 1360161 | doi = 10.1126/science.1105891 | bibcode = 2005Sci...309.2010R }}</ref> and small synthetic genetic networks,<ref>{{cite journal | vauthors = Elowitz MB, Leibler S | title = A synthetic oscillatory network of transcriptional regulators | journal = Nature | volume = 403 | issue = 6767 | pages = 335–338 | date = January 2000 | pmid = 10659856 | doi = 10.1038/35002125 | s2cid = 41632754 | bibcode = 2000Natur.403..335E }}</ref><ref>{{cite journal | vauthors = Gardner TS, Cantor CR, Collins JJ | title = Construction of a genetic toggle switch in Escherichia coli | journal = Nature | volume = 403 | issue = 6767 | pages = 339–342 | date = January 2000 | pmid = 10659857 | doi = 10.1038/35002131 | s2cid = 345059 | bibcode = 2000Natur.403..339G }}</ref> such as the genetic toggle switch of Tim Gardner and [[James Collins (bioengineer)|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]].<ref>{{cite journal |author=Gillespie DT |title=A general method for numerically simulating the stochastic time evolution of coupled chemical reactions |journal=J. Comput. Phys. |volume=22 |pages=403–34 |year=1976 |doi=10.1016/0021-9991(76)90041-3 |issue=4|bibcode=1976JCoPh..22..403G }}</ref>

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.<ref>{{cite journal | vauthors = Roussel MR, Zhu R | title = Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression | journal = Physical Biology | volume = 3 | issue = 4 | pages = 274–284 | date = December 2006 | pmid = 17200603 | doi = 10.1088/1478-3975/3/4/005 | bibcode = 2006PhBio...3..274R | s2cid = 21456299 }}</ref>

From here, a set of reactions were proposed<ref>{{cite journal | vauthors = Ribeiro A, Zhu R, Kauffman SA | title = A general modeling strategy for gene regulatory networks with stochastic dynamics | journal = Journal of Computational Biology | volume = 13 | issue = 9 | pages = 1630–1639 | date = November 2006 | pmid = 17147485 | doi = 10.1089/cmb.2006.13.1630 | s2cid = 6629364 }}</ref> 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<sub>&nbsp;''i''</sub> is the promoter region of gene ''i''):

: <math> \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) </math>

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.<ref name="a30">{{cite journal | vauthors = Zabet NR, Chu DF | title = Computational limits to binary genes | journal = Journal of the Royal Society, Interface | volume = 7 | issue = 47 | pages = 945–954 | date = June 2010 | pmid = 20007173 | pmc = 2871807 | doi = 10.1098/rsif.2009.0474 }}</ref><ref name="a31">{{cite journal | vauthors = Chu DF, Zabet NR, Hone AN | title = Optimal parameter settings for information processing in gene regulatory networks | journal = Bio Systems | volume = 104 | issue = 2–3 | pages = 99–108 | date = May–Jun 2011 | pmid = 21256918 | doi = 10.1016/j.biosystems.2011.01.006 | bibcode = 2011BiSys.104...99C | url = https://kar.kent.ac.uk/30778/1/optimalParameterSettings_Chu.pdf }}</ref><ref name="a32">{{cite journal | vauthors = Zabet NR | title = Negative feedback and physical limits of genes | journal = Journal of Theoretical Biology | volume = 284 | issue = 1 | pages = 82–91 | date = September 2011 | pmid = 21723295 | doi = 10.1016/j.jtbi.2011.06.021 | arxiv = 1408.1869 | s2cid = 14274912 | bibcode = 2011JThBi.284...82Z }}</ref>

A recent work proposed a simulator (SGNSim, ''Stochastic Gene Networks Simulator''),<ref>{{cite journal | vauthors = Ribeiro AS, Lloyd-Price J | title = SGN Sim, a stochastic genetic networks simulator | journal = Bioinformatics | volume = 23 | issue = 6 | pages = 777–779 | date = March 2007 | pmid = 17267430 | doi = 10.1093/bioinformatics/btm004 | doi-access = free }}</ref> 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.<ref>{{cite journal | vauthors = Kaznessis YN | title = Models for synthetic biology | journal = BMC Systems Biology | volume = 1 | pages = 47 | date = November 2007 | pmid = 17986347 | pmc = 2194732 | doi = 10.1186/1752-0509-1-47 | doi-access = free }}</ref>