Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Signaling Pathways, Modeling of

  • Jeanette Hellgren KotaleskiEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_195-1


Enzymatic Reaction Dendritic Spine Biochemical Reaction Deterministic Approach Reaction Cascade 
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The use of models to study the dynamical behavior of a biological (sub)system described as a set of biochemical reactions and diffusion. Here, the evolution in time for quantities such as protein concentrations or amount of enzyme activation is typically described using ordinary differential equations. The evolution in time of second messengers is typically described using partial differential equations. Alternatively, stochastic methods can be used to determine evolution in time of all molecules in a simulation.

Detailed Description

The development of quantitative models at multiple spatial and temporal scales is necessary for integrating the knowledge obtained from diverse experimental approaches into a coherent picture. Such models represent current knowledge in a compact and standardized way and constitute a tool for guiding experiments and generating predictions.

Modeling of intracellular signaling within the field of computational neuroscience is typically done to better understand how events on the subcellular level affect, or even may explain, higher-level phenomena. For example, subcellular mechanisms underlying synaptic long-term potentiation (LTP) and long-term depression (LTD) could be investigated in models of receptor-induced cascades of signaling molecules activated during typical experimental protocols. Commonly modeled LTP or LTD systems begin with the activation of transmembrane receptors or the influx of calcium and include the activation of kinases and/or phosphatases and their phosphoprotein targets (see, e.g., Kotaleski and Blackwell 2010).

Dynamical modeling of signaling pathways is required because steady states are rarely achieved. Most of the time activation of signaling pathways is transient, and thus, the response of the system is transient. Intracellular signaling pathways may reach a steady state solution if the activation is kept constant, but sometimes such systems can show oscillations or behave as a bistable (or even multistable) biochemical switch.

How to Represent the Reactions in the Signaling Pathway?

Signaling pathways are represented as cascades of biochemical reactions, together with the diffusion of a subset of the molecular species. It is not plausible to include every reaction known to occur in neurons or cells; thus, the modeled pathway includes those components, such as enzymes, proteins, calcium ions, etc, judged to be relevant for the research question. The identification of critical components and how they interact or react typically are based on experimental data. Such data is often qualitative, but quantitative data are required for model development. Once the relevant molecules and reactions have been identified, the system can be specified and simulated. Two general approaches to the simulations are stochastic and deterministic. When using a deterministic approach, the system of biochemical reactions are described using a system of ordinary differential equations, which describe the rate of change in the concentrations/amounts of each component. The rate of change is based on the concentrations of all other reactants involved in a reaction. In addition, the diffusion of molecules is represented using discretized versions of partial differential equations. Various software packages that include numerical integrators can then be used to simulate the dynamics of the system if initial conditions and reaction rate parameters are provided.

To capture the individual steps in a signaling pathway, bimolecular and enzymatic reactions as well as protein modifications need to be included. Simple biochemical reactions, where one reacting substance (A) binds to another one (B) to form a product (AB) or to modify each other (A*, B*), can be represented as
$$ \mathrm{A}+\mathrm{B}\leftrightarrow \mathrm{A}\mathrm{B},\mathrm{or} $$
$$ \mathrm{A}+\mathrm{B}\leftrightarrow {\mathrm{A}}^{*}+{\mathrm{B}}^{*} $$

Here, it is assumed that in both these cases, the reaction can be reversed. The speed of the forward and backward reactions is described by corresponding reaction rate parameters, kf and kb.

The “A + B ↔ AB” system in Eq. 1, for example, is modeled deterministically using the following first-order differential equations:
$$ \mathrm{d}\left[\mathrm{A}\right]/\mathrm{dt}=\mathrm{d}\left[\mathrm{B}\right]/\mathrm{dt}=-{\mathrm{k}}_{\mathrm{f}}\cdotp \left[\mathrm{A}\right]\cdotp \left[\mathrm{B}\right]+{\mathrm{k}}_{\mathrm{b}}\cdotp \left[\mathrm{A}\mathrm{B}\right]\ \mathrm{and} $$
$$ \mathrm{d}\left[\mathrm{A}\mathrm{B}\right]/\mathrm{dt}={\mathrm{k}}_{\mathrm{f}}\cdotp \left[\mathrm{A}\right]\cdotp \left[\mathrm{B}\right]-{\mathrm{k}}_{\mathrm{b}}\cdotp \left[\mathrm{A}\mathrm{B}\right], $$
where [A], [B], and [AB] stand for the concentrations (e.g., measured in μM) of the corresponding reactants or products. The kf and kb rate parameters have here units of concentration−1 time−1 and time−1, respectively. The units for concentration of molecules must match the units of the rate constants. One additional set of values are required: given the initial concentrations for A, B, and AB, one can numerically integrate the behavior of the system over time.
When the system Eq. 1 above has reached steady state, ss, so that all time derivatives of the concentrations are zero, then
$$ {\mathrm{k}}_{\mathrm{b}}/{\mathrm{k}}_{\mathrm{f}}=\left[{\mathrm{A}}_{\mathrm{ss}}\right]\cdotp \left[{\mathrm{B}}_{\mathrm{ss}}\right]/\left[{\mathrm{A}\mathrm{B}}_{\mathrm{ss}}\right], $$
where kb/kf defines the dissociation constant, KD, which thus can be estimated from the steady state concentrations of the involved species. (Note that sometimes the affinity constant, KA = 1/KD, is given instead.) Measures of KD are provided in the biochemical literature more frequently than individual rate constants, because to estimate kb and kf individually requires data on the dynamics in the system, which are difficult experiments to perform.

The above approach assumes mass action kinetics which states that the rate of a reaction is the product of a rate constant (k) times the concentrations of the reacting species. This approach can be used to describe the behavior of both reversible and irreversible reactions, also when these reactions are part of reaction cascades, such as simple enzymatic reactions, as explained below. A similar framework, A ↔ A*, can also represent the exchange of mass through diffusion between different compartments (dendritic spines, dendritic shaft, or different subvolumes within a spine or a dendrite). The rate by which the species move between compartments can be represented by the diffusion coefficient adjusted for the volume and contact areas of the adjacent compartments.

Mass action kinetics grew out of the work by Guldberg and Waage (reviewed by Lund 1965) who investigated, and tried to find laws for, how the velocity of chemical reactions could be described. This law describes how fast the molecules in a well-stirred solution collide and interact on average (and thus, stochastic interactions between individual species molecules are ignored).

Enzymatic reactions can be modeled as a cascade of biochemical reactions with the approach above. Enzymes speed up the rate by which the substrate of the enzyme is transformed into a product. The underlying reactions are assumed to be captured by the following scheme, which in the simplest case assumes a nonreversible catalytic step, a scheme developed successively in the early twentieth century by Michaelis and Menten (1913) and Briggs and Haldane (1925):
$$ \mathrm{E}+\mathrm{S}\leftrightarrow \mathrm{E}\mathrm{S}\to \mathrm{E}+\mathrm{P} $$
In the first step, described as above by a kf and kb rate parameter, the enzyme (E) binds to the substrate (S) in an enzyme-substrate compound (ES). The second reaction step, where the enzyme is catalyzing the formation of the product, is controlled by the catalytic rate parameter, kcat. Again, given some initial conditions and some values on the rate parameters, kf, kb, and kcat, these two reaction steps can be modeled deterministically using first-order differential equations of the form
$$ \mathrm{d}\left[\mathrm{E}\right]/\mathrm{dt}=-{\mathrm{k}}_{\mathrm{f}}\cdotp \left[\mathrm{E}\right]\cdotp \left[\mathrm{S}\right]+{\mathrm{k}}_{\mathrm{b}}\cdotp \left[\mathrm{E}\mathrm{S}\right]+{\mathrm{k}}_{\mathrm{cat}}\cdotp \left[\mathrm{E}\mathrm{S}\right] $$
$$ \mathrm{d}\left[\mathrm{S}\right]/\mathrm{dt}=-{\mathrm{k}}_{\mathrm{f}}\cdotp \left[\mathrm{E}\right]\cdotp \left[\mathrm{S}\right]+{\mathrm{k}}_{\mathrm{b}}\cdotp \left[\mathrm{E}\mathrm{S}\right] $$
$$ \mathrm{d}\left[\mathrm{E}\mathrm{S}\right]/\mathrm{dt}={\mathrm{k}}_{\mathrm{f}}\cdotp \left[\mathrm{E}\right]\cdotp \left[\mathrm{S}\right]-{\mathrm{k}}_{\mathrm{b}}\cdotp \left[\mathrm{E}\mathrm{S}\right]-{\mathrm{k}}_{\mathrm{cat}}\cdotp \left[\mathrm{E}\mathrm{S}\right] $$
$$ \mathrm{d}\left[\mathrm{P}\right]/\mathrm{dt}={\mathrm{k}}_{\mathrm{cat}}\cdotp \left[\mathrm{ES}\right] $$

A common example is the reaction catalyzed by a kinase, which phosphorylates a substrate, S + ATP → P + ADP, where the substrate, S, is a protein to be phosphorylated, P. This simple, one-step reaction assumes that ATP is constant (and sufficiently high that it is not rate limiting) and all substrate molecules have formed a complex with the enzyme. Then, the reaction rate is captured by the kcat rate constant above. In theory, an enzymatic reaction must be reversible since an enzyme just speeds up the reaction, but does not change the steady state, which is decided by the thermodynamics constraints. Thus, if the product is accumulating in high enough amounts, a reversible reaction may need to be accounted for. Whether this needs to be represented in a model depends on the equilibrium constants for the enzymatic reactions considered.

Enzymatic reactions are sometimes modeled deterministically using the Michaelis-Menten kinetics. With this simplification, the variations in [E] and [ES] are discarded from Eq. 6ad and instead assumed to be constant, i.e., d[ES]/dt = 0, as during steady state conditions. Since the amounts of the substrate and product change over time, this assumption requires that the substrate be present in sufficient amount, else the amount of [ES] will change over time. Whether the enzymatic reaction can be simplified to the Michaelis-Menten (MM) formalism or needs to be modeled explicitly using Eq. 6 depends therefore on the particular system modeled.

When modeling a set of signaling pathways as a cascade of biochemical reactions, the product of one reaction is usually the substrate of a subsequent reaction. Alternatively, active enzymes are often the product of a reaction. Frequently, enzyme and bimolecular reactions alternate. For example, first a ligand binds to a receptor; second, the ligand receptor complex is an enzyme; third, product of the enzyme reaction binds to and activates another enzyme, etc. When the product of the enzyme is a diffusible molecule, it is often called a second messenger. The diffusion and degradation of the second messenger must be modeled carefully to correctly represent its spatial and temporal dynamics.

When searching for parameter values in the literature for the various enzymatic reactions to be represented in the model, usually only kcat (or more often Vmax which is kcat · total amount of enzyme) and the concentration of substrate when the production of product is half-maximal, KM, is reported. Thus, the kf, kb, and kcat in the equations above need to be mapped onto KM, and it can be shown that (kb + kcat)/kf = KM. This, however, means that we only have two measured values but three parameters to set. Thus, when kb is not known explicitly, it needs to be set, e.g., as 4 · kcat as motivated in Bhalla and Iyengar (1999). Under the assumption that d[ES]/dt = 0, and setting [E] = [Etotal] − [ES], then
$$ \mathrm{d}\left[\mathrm{P}\right]/\mathrm{dt}={\mathrm{k}}_{\mathrm{cat}}\cdot \left[\mathrm{Etotal}\right]\cdot \left[\mathrm{S}\right]/\left({\mathrm{K}}_{\mathrm{M}}+\left[\mathrm{S}\right]\right) $$

Whether the Michaelis-Menten (MM) representation, Eq. 7, and the explicit formulation above, Eqs. 6a6d, give similar results or not depends on to what extent the assumptions underlying the MM formalism are fulfilled. In, e.g., Ramakrishnan and Bhalla (2008), there is one example of a minimal biochemical system that can behave as a switch when the system is modeled using the MM formalism, but not when the explicit formulation above is used.


Although the above deterministic approaches are commonly used within the computational neuroscience field (Manninen et al. 2010), several factors should be considered when representing a signaling pathway with a set of ordinary differential equations, assuming well-stirred solutions. For example, receptor-induced cascades involved in synaptic plasticity typically are localized in dendritic spines, and compartmentalization with regard to functions of different molecules might be common. Also proteins are heterogeneously distributed in different parts of the cytosol and membrane, and some molecules are anchored, while others might diffuse freely. Modeling these reactions across different compartments requires diffusion. If reacting or diffusing species are present in small quantities, then reactions and diffusion occur stochastically, in which case they cannot be modeled deterministically, and stochastic approaches must be used.

One overall challenge when building models of signaling pathways is the estimation of the reaction rate constants and amounts of the different molecular species. Biochemical estimates of model parameters can, for instance, be obtained through “test tube” experiments. However, the conditions in real cells might vary significantly from those in the experiments.

Although one often talks about a signaling pathway, it should be realized that such a reaction cascade is most likely embedded in a larger signaling network and that many interactions from other parts of the bigger system are left out. This also makes it difficult to validate pathways models, e.g., a pathway leading from a certain G-protein-coupled receptor to the activation of an enzyme, if those pathways are present in different cell types. Although the reaction rates between the pathway’s individual components are assumed to be similar, differences in the left-out interacting network might affect effective concentrations of enzymes and substrates through sequestration effects, for example, through the competition for available pools of enzymes and substrates.


  1. Bhalla US, Iyengar R (1999) Emergent properties of networks of biological signaling pathways. Science 283(5400):381–387PubMedCrossRefGoogle Scholar
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  3. Kotaleski JH, Blackwell KT (2010) Modelling the molecular mechanisms of synaptic plasticity using systems biology approaches. Nat Rev Neurosci 11:239–251PubMedCrossRefGoogle Scholar
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Further Reading

  1. Cornish-Bowden A (2012) Fundamentals of enzyme kinetics, 4th edn. Wiley-Blackwell, Weinheim, ISBN 978-3-527-33074-4Google Scholar
  2. Johnson KA, Goody RS (2011) The original Michaelis constant: translation of the 1913 Michaelis-Menten paper. Biochemistry 50:8264–8269PubMedCentralPubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Computer Science and CommunicationKTH Royal Institute of TechnologyStockholmSweden