Basic Concepts in Modelling Biological Systems

To start with, we look at conceptual representation of biological systems in schematic diagrams and then review some of the basic chemical reaction laws and formulation for: Mass Action kinetics, Michaelis-Menten kinetics, Michaelis-Menten competitive inhibition and Hill function.

3.2.1

Schematic Diagram of a Biological System

A schematic diagram can be used to describe the molecular interactions in a biological system based on the known experimental findings and is “an important first step” in representing and

understanding a system’s structure and dynamics (Kitano, 2002a). It captures the essential components that play an important role in the biological system. It is typically used in this field to illustrate a complex model in a simple way because a diagram can help reader understand the concept or interactions much more easily and quickly. A schematic diagram or wiring diagram is also used to describe the model hypothesis that can then transform into mathematical model equations. Schematic diagrams have been widely used to present model hypotheses that lead to the

construction of kinetic model equations. For this reason, it is very crucial to draw a clear schematic diagram for our biological system of interest and present a clear picture of the problem and model hypothesis for the molecular mechanism that we are investigating. In the next section (Section 3.3), we illustrate how a schematic diagram is used to represent the molecular mechanism that leads to

the construction of model equations using the p53 model from Geva-Zatorsky et al. (2006) as an example. Before that let us look at the basic kinetic laws in modelling chemical reactions.

3.2.2

Kinetic Modelling of Chemical Reactions

In this section, a review is given on some of the essential laws for kinetic modelling of chemical reactions that will be used in formulating ordinary differential equations. There are four kinetics laws: 1) Law of Mass Action, 2) Michaelis-Menten, 3) Michaelis-Menten competitive inhibition and 4) Hill function. Cells form complex networks of interacting macromolecules such as DNAs, mRNAs and proteins. These networks can be modelled as a set of chemical reactions that involve substrates (S) being converted to products (P) by enzymes (or proteins). An enzyme (E) acts as a catalyst that accelerates the rate of a reaction.

1. Law of Mass Action

Let us consider a reversible reaction below:

The mass action kinetics states that the rates of reaction are proportional to the concentrations of the reactants (Aldridge et al., 2006). Therefore, we assume the rate of forward reaction is linearly proportional to the concentrations of A and B, and the backward reaction is linearly proportional to the concentration of C. Thus, the rate equations are:

𝒅𝒅[𝑨𝑨] 𝒅𝒅𝒅𝒅 =𝒌𝒌−𝟏𝟏[𝑪𝑪]− 𝒌𝒌𝟏𝟏[𝑨𝑨][𝑩𝑩] (3.1) 𝒅𝒅[𝑩𝑩] 𝒅𝒅𝒅𝒅 =𝒌𝒌−𝟏𝟏[𝑪𝑪]− 𝒌𝒌𝟏𝟏[𝑨𝑨][𝑩𝑩] (3.2) 𝒅𝒅[𝑪𝑪] 𝒅𝒅𝒅𝒅 =𝒌𝒌𝟏𝟏[𝑨𝑨][𝑩𝑩]− 𝒌𝒌−𝟏𝟏[𝑪𝑪] (3.3)

where [A], [B] and [C] represent the concentration for molecular species of A, B and C, respectively.

A

+ B

k

C

k-

2. Michaelis-Menten Kinetics

For enzyme-catalysed reaction, we consider a reaction given below:

The Michaelis-Menten mechanism for an enzyme-catalysed reaction: E binds to the substrate S to form an enzyme-substrate complex ES; in the complex, E converts S to P; once the conversion is done, E dissociates from P and is free to bind another molecule of substrate (Conrad & Tyson, 2006). Assuming that the total enzyme concentration ET is much less than the initial substrate concentration

S0, the rate of the enzyme-catalysed reaction is given by:

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 =𝑘𝑘2

[𝐸𝐸𝑇𝑇][𝑆𝑆]

𝐾𝐾𝑚𝑚+[𝑆𝑆] (3.4) where _{𝐾𝐾𝑚𝑚=}𝑘𝑘−1+𝑘𝑘2

𝑘𝑘1 is called the Michaelis constant and k2 is a rate constant. The Michaelis-Menten formula is also commonly expressed as:

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑 =𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚

[𝑆𝑆]

𝐾𝐾𝑚𝑚+[𝑆𝑆] (3.5) where Vmax = k2[ET].

3. The Michaelis-Menten Competitive Inhibition Kinetics

For enzyme-catalysed reaction (with competitive inhibition), we consider a reaction given below:

The process is almost the same as the above Michaelis-Menten mechanism for an enzyme-catalysed reaction: E binds to the substrate S to form an enzyme-substrate complex ES; in the complex, E converts S to P; once the conversion is done, E dissociates from P and is free to bind another molecule of substrate. However, in addition, there is an inhibitor I binding to E to form EI in a reversible reaction.

S

+ E

k

ES

k-

k

_{P + E}

Enzyme-substrate

complex

S

ES

+

E

k

k-

k

_{P + E}

Enzyme-substrate

complex

+

I

EI

K

_K

The Michaelis-Menten equation for competitive inhibition (Klipp et al., 2008) is expressed as:

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑 =𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚

[𝑆𝑆]

𝐾𝐾𝑚𝑚(1+[_{𝐾𝐾𝑖𝑖}𝐼𝐼])+[𝑆𝑆] (3.6) where [I] is the concentration of inhibitor.

4. Hill Function

A reaction can bind more than one molecule from a given substrate. Usually, binding of the first substrate molecule changes the rate at which the second substrate molecule binds. If the binding rate of the second substrate molecule is increased, it is called positive cooperativity. This property of positive cooperativity is approximated by a Hill function (Klipp et al., 2008) given below:

𝑓𝑓(𝑥𝑥) =𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚

𝑛𝑛

𝐾𝐾𝑛𝑛_+𝑚𝑚𝑛𝑛 (3.7) where n is defined as Hill coefficient and n more than one indicates cooperative binding. Usually, n is assumed to be a positive integer such as 1, 2, 3 or 4. (Note: when n=1, it gives the Michaelis-Menten formula)