Interlude: kinetic modelling in a nutshell

One widely used approach for deriving equations that describe how components of a system change over time is called kinetic modeling. Although kinetic modeling has its roots in (bio)chemical reaction systems, it also serves as the foundation for models across many scales — including ecology, epidemiology, and systems biology.

Kinetic models represent processes as sets of biochemical reactions, where substrates (usually one or two) are transformed into products. The speed at which each reaction occurs, known as the reaction rate, determines how quickly substrates are consumed and products are formed.

For each component in the system, its rate of change is given by the sum of the contributions from all reactions in which it participates, either as a substrate or a product. Each contribution is calculated by multiplying the reaction rate by the stoichiometric coefficient of that component in the respective reaction. For more details and a nicely worked example you can consult: How and why to build a mathematical model: A case study using prion aggregation (Banwarth-Kuhn M and Sindi S, 2020).

An example of a general reaction

To make this concrete, consider the following generic biochemical reaction:

Where:

• A and B are model components that act as substrates,
• C and D are model components that act as products,
• a and b are the stoichiometric coefficients of the substrates A and  B, 
• c and d are the stoichiometric coefficients of the products C and D and
• k is the rate constant.

This reaction could describe a step in a process where a molecules of A and b molecules of B bind to each other and are transformed into c molecules of C and d molecules of D.

Reaction Rate: Mass Action kinetic rate law

What is happening in a reaction is one thing, how fast the reaction is happening is another.  So called rate laws are used to quantify the speed of the reaction, the reaction rate, by relating it to for example the concentrations of the substrates.  One commonly used rate law is the mass action rate law. It states that the reaction rate is proportional to the product of the concentrations of the substrates, raised to the power of their stoichiometric coefficients, the proportionality factor being the rate constant k. For our generic reaction from above that would be:

Where:

• r is the reaction rate,
  
• k is the rate constant,
  
• [A] and [B] are the concentrations of A and B.

Note: The mass action kinetic rate law derives the rate of a reaction. In the context of chemical reactions it is only correct for elementary chemical reactions. The related law of mass action, however, is about the relationship between product and substrate concentrations at chemical equilibrium and is valid for all chemical reactions no matter in how many steps they proceed. The statement that mass action kinetics are a direct result from, or dictated by, the law of mass action is a common misconception. The law of mass action does not state anything about reaction kinetics.

Rate of Change of Components

Now that we know how fast the reaction happens, we can write down the equation for how the concentration of each reaction component changes over time.

For A, a substrate being consumed the change is given by:

f_A ​= − ak [A]^a [B]^b

This term is negative because A is used up by the reaction.

For C, a product being formed change is given by:

f_C = ck [A]^a [B]^b

This term is positive because C is produced by the reaction.

If component A or C are involved in multiple reactions, their total rate of change will  be the sum of the reaction rates from each reaction, each weighted by the respective stoichiometric coefficient of the component in the respective reaction.