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Biochemical Reaction Networks: A Review
Features of Dynamical Models in Systems Biology
We have previously detailed the crucial role that quantitative mathematical (or dynamical) models plays in modern biological reasoning. Herein, we elaborate on their required components.
The primary components of a dynamical mathematical model of a biological system corre-spond to the molecular species present in the system.The abundance of each species is assigned to a state variable of the model, the collection of which represents the system state. The system state provides a complete description of the system’s condition at any given time, via its time course, as given by the model’s dynamic behavior.
Besides state variables, models of biological systems also include parameters, which charac-terize environmental effects and interactions among system components, and whose values are fixed – meaning that the distinction between model parameters and state variables is clear-cut. Example of common parameters include association constants, maximal expression rates, and degradation rates. As a change in the value of a model parameter corresponds to a change in the environmental conditions or in the system itself, model parameters are typically held con-stant during a simulation. Varying parameter values between rounds of simulation allows one to explore system behavior under perturbations of the experimental conditions, or in altered environments.
As the main feature of a dynamical model of a biological system is its ability to describe the temporal evolution of the system components’ quantities, building a model then involves two important choices: (i) how to represent the model structure (i.e., determine its species, parameters, and molecular interactions), which is akin to defining its syntax, and (ii) how to ’interpret’, or execute, the model, which is akin to defining its semantics. These two steps are indicative of a distinction that is commonly made when dealing with systems modeling: the qualitative approach, respectively the quantitative approach. While the latter essentially relies on mathematical equations describing the performance of the system for a large set of input functions and initial states, the former requires no such formal mathematical formulation, instead relying on visual representations (e.g., diagrams) of the relationships between the system components, that is, it relies on the structural model.
For a large class of biological systems, what one usually seeks to model are the interactions between individual molecules which are only distinguishable by the class of species they belong to. Consequently, population models prove to be a well-suited, and popular, mathematical modelling framework for biological phenomena. Any population model can be described as a set of reactions operating on a set of biochemical species: this is what we will refer to as Biochemical Reaction Networks; they will represent the modeling framework of choice throughout this thesis. Below, we motivate our choice.
Network models of biological systems
While using mathematical models in the natural sciences (particularly in physics) is not a new idea1, the novelty lays in the use of such models in life sciences, most notably in biology and biomedicine. As such, quantitative mathematical models have lain outside mainstream research approaches during the last decades of the purposely reductionist qualitative era of molecular biology, which instead heavily employed qualitative models. Indeed, biologists have long used these type of models as abstractions of reality, be it in the familiar form of the ball-and-stick model of chemical structure, or more generally in the form of diagrams that illustrate a set of components and their interactions, and which play a central role in representing our knowledge of cellular processes [99].
Figure 1.1: An interaction diagram, inspired from [99]. Species A and B bind reversibly, forming a complex that inhibits the rate at which species C is transformed into species D. The graphical conventions used are that the blunt arrows denote inhibition, normal arrows denote activation, while dashed lines denote regulatory interactions, i.e. the species is not consumed by the reaction.
Such interaction maps, also called networks, formalize the complex interactions between heterogeneous entities (within and between cells), which characterize biological systems. For example, Figure 1.1 shows molecular species A and B binding reversibly, to form a complex that inhibits the rate at which species C is transformed into species D.
Generally speaking, such network-based modeling approaches help structure and formalize existing knowledge, as well as predict system behaviour, and have been widely used for describ-ing cellular regulation and metabolism [16]. Consequently, it makes sense that a quantitative modeling framework be developed using this widely-employed qualitative description of biologi-cal systems: enter Biochemical Reaction Networks (BRNs). The central argument that justifies adding quantitative/mathematical information to simple interaction maps, in order to obtain quantitative BRN models, is that simple interaction diagrams leave ambiguities with respect to the system’s behaviour: we know what components of the system interact with each other, but not necessarily how. By employing a mathematical description of the system, this uncertainty can be eliminated, at the cost of demanding a quantitative characterization of every interaction depicted in the diagram [99].
For example, in order to quantify the interaction between molecular species A and B, in Figure 1.1, a numerical description of the process must be provided, under the form of the bind-ing and unbinding reaction rate constants. For cellular processes of which only a qualitative understanding of the underlying molecular interactions is available, such a quantitative descrip-tion is not possible. However, for an important number of well-studied mechanisms, sufficient data have been collected as to allow a quantitative characterization [99], that, when coupled with the interaction diagram, can be used to formulate a mathematical model of the network’s dynamics, which typically involves the physical and chemical laws governing the mechanisms that drive the observed behaviour (such models are dubbed mechanistic). Whenever the quan-titative information regarding molecular interactions is available, the result is a collection of biochemical reactions involving the system’s species. A species is a system entity that is quan-tifiable, and whose quantity is susceptible of evolving over time. A reaction is an elementary action of the system that consists of consuming (respectively producing) a finite and integer quantity of a finite subset of species, called reactants(respectively products).
For example, the reversible binding reaction between species A and B of the interaction diagram in Figure 1.1 writes as:
We note that, at this level of modeling, the notation A.B for the complex species is simply a name, and should not be considered as indicative of the reaction’s mechanistic details (i.e., A binds B). In this sense, one could choose any other name for the complex species: e.g., reaction.
The model parameters are the binding, respectively unbinding, reaction constants κ1 and κ−1. As we will see in this chapter, both the interpretation of the reaction constants and the representation of the system state will depend on the chemical kinetics assumed to govern the dynamics of the system.
Biochemical Reaction Networks will be the modeling framework of choice throughout this thesis. Consequently, in the next sections of this chapter we formally define both their syn-tax, as well as the two major applicable semantics: classical (deterministic) chemical kinetics, respectively stochastic chemical kinetics.
Before moving forward with the definition of BRNs, we note that a disadvantage of network models is that as the number of components and interactions in a biological system grows, it becomes increasingly difficult to maintain an intuitive understanding of the overall behaviour
[99]. Thus, different levels of information abstraction can be employed when constructing network models, according to the size of the system, the available data, and the research question under consideration. In their simplest form, the interaction maps only depict the possibility of interaction between genes or proteins (Figure 1.2, top) – and can therefore cover larger systems -, while at the other end of the scale one finds detailed models based on mathematical descriptions (Figure 1.2, bottom), which provide significantly more detailed insight into the dynamics, but are usually only used to describe small, well-studied subsystems. Figure 1.2 shows a hierarchy of different, widely-employed, network models, each one abstracting information on a different level, and thus requiring different amounts of detail.
Figure 1.2: A hierarchy of network models for biological systems [16]. Existing net-work models of biological systems abstract information on different levels, depending on the available data and the biological questions under consideration. Top-down: depicted model types are ranked in increasing detail level and data demand, and decreasing coverage power. Less detailed models cover larger systems (in number of genes/biomolecules), but do not allow for true simulation of network dynamics – simulation is possible only for the most detailed cat-egory of models, dubbed “quantitative models”, that are labeled with kinetic rate constants. Intermediary models, i.e. causal and logical networks, allow at most for simulation of the event sequence or gene activation order, but quantifying the speed of reactions is impossible under these simplified frameworks.
Biochemical Reaction Networks: Syntax
Models of cellular phenomena often take the form of interaction diagrams, as in Fig.1.1. For biochemical and genetic networks, interaction diagrams depict the molecular species in the system – which could be ions, small molecules, macromolecules, or molecular complexes – as nodes, and the interactions between them as arrows. The arrows can represent a range of processes: chemical binding or unbinding, reaction catalysis, or regulation of activity. The processes result in the production, inter-conversion, transport, or consumption of the species within the network.
A set of reactions constitutes a biochemical reaction network. The manner in which the biochemical species interact is referred to as the network topology, and its organization is ap-parent if one rearranges the reactions in the form of an interaction graph. For example, the interaction graph of the Michaelis-Menten mechanism of Ex.1.2.1 is shown in Fig.2.1.
In order to obtain a quantitative description, one needs to know the rates at which the reactions occur. In vivo, the rate of a reaction depends on its kinetic constant, on the con-centration of the reactants, and on physico-chemical conditions, such as temperature and pH. However, for in silico models, the physico-chemical conditions are fixed, such that rate laws can be described solely in terms of reactant concentrations and the kinetic rate constants.
Table of contents :
I Biochemical Reaction Networks: A Review
1 Context and Motivation
1.1 Features of Dynamical Models in Systems Biology
1.2 Network models of biological systems
2 Biochemical Reaction Networks: Syntax
3 Biochemical Reaction Networks: Semantics
3.1 Classical chemical kinetics
3.2 Stochastic chemical kinetics
3.3 Stochastic vs deterministic models
4 Executable Biology: Computational models of Biochemical Reaction Networks
4.1 Petri Nets
4.1.1 Motivation
4.1.2 Qualitative Petri Nets
4.1.3 Quantitative Petri nets
4.2 Rule-based modeling and the Kappa language
4.2.1 Rule-based modeling
4.2.2 The Kappa language: Syntax and Operational Semantics
4.2.3 The Kappa language: Stochastic Semantics
II Model reduction
5 Prototyping genetic circuits using rule-based models
5.1 Motivation
5.2 A Kappa model of the -phage decision circuit
5.2.1 Example: Kappa model of CI and CII production
5.3 Model Approximation using Michaelis-Menten-like reaction schemes
5.3.1 Validity of the Michaelis-Menten enzymatic reduction in stochastic models
5.3.2 Generalized competitive enzymatic reduction
5.3.3 Operator site reduction
5.3.4 Fast dimerization reduction
5.3.5 Top-level reduction algorithm
5.4 Results and Discussion
5.4.1 Results: reduction of the -phage decision circuit
5.4.2 Conclusion and future work
6 Tropical Abstraction of Biochemical Reaction networks with guarantees
6.1 Introduction
6.2 Definitions and Motivating Examples
6.2.1 General Setting and Definitions
6.2.2 Motivating example: Michaelis-Menten
6.2.3 Motivating example: A DNA model
6.3 Model reduction using conservative numerical approximations
6.4 Error estimates of tropicalized systems using conservative numerical approximations
6.4.1 Tyson’s Cell Cycle Model
6.5 Comparison with existing methods
6.6 Conclusion and outlook
III Modelling storage and growth
7 Effects of cellular resource storage on growth
7.1 Introduction
7.2 Review of the Weisse cell model
7.2.1 Overview
7.2.2 Nutrient Uptake and Metabolism
7.2.3 Transcription
7.2.4 Competitive Binding
7.2.5 Translation
7.2.6 Growth and cellular mass
7.2.7 Model parameters
7.3 A scaling procedure for modelling nutrient storage in the cell
7.3.1 Motivation
7.3.2 Definition
7.3.3 Examples
7.3.4 Storage capacity in the Weisse model
7.4 The Growth Rate is an emergent property of the model
7.5 The Single Cell model
7.5.1 The effects of metabolite storage on exponential growth rate
7.5.2 Metabolite storage and adaptation to environmental fluctuations
7.5.3 Environmental dynamics and resource storage strategies
7.6 The Ecological perspective
7.6.1 Motivation
7.6.2 Experiment setup: a mixed population model in a fluctuating environment174
7.6.3 Results
7.7 Conclusion
8 Abstractions for cellular growth: towards a new Petri net semantics
8.1 Motivation
8.2 Split-Burst: towards a piecewise-synchronous execution of Biochemical Reaction Networks
8.2.1 Max-parallel execution semantics of Petri Nets
8.2.2 Piecewise-synchronous execution semantics
8.3 Linear Reaction Networks
8.3.1 Growth rate as an emergent property
8.3.2 Synchronous execution
8.4 Possible Applications
8.4.1 Approximation of system dynamics
8.4.2 Synchronous Balanced Analysis
8.5 Conclusion and future directions
9 Conclusion and future directions
9.1 Contributions
9.2 Future works
