# Modeling intracellular interactions between sender and receiver bacteria

I am involved in a modelling task of a biological process described as follows:

-- "Objective: to model behavior of a sender receiver type system

The process goes as follows

• Pre-signal [PS] is in media

• Pre-signal diffuses into bacteria

• Pre-signal binds to activator as a co-activator

• Co-activator activator complex binds to DNA and starts transcription of enzyme mRNA

• Enzyme mRNA is transcribed then translated into enzyme

• Enzyme begins converting Pre-signal to signal [S]

• Signal diffuses out of the cell into media

• Signal diffuses into receiver cell from media

• Signal binds to repressor inducing activation of reporter gene

Questions for model:

What concentrations of signal will be required to get a response from receiver cell?"

--

My questions are: what kind of mathematical/computational model or simulation tool is more suitable for this task? is it neccesary to model every step, or can some of these be ignored? I know the basics of ODEs, so I wonder if these are enough or something else would be required. Any other advice about how to go through the modelling is welcome.

However, if you are considering diffusion then you would have to use Partial differential equations for Fick's diffusion law.

You can, however, assume that there is no "diffusion" but just transport of the signal molecule from a well mixed system (medium) to another well mixed system (cytosol). In this case you can use an ODE for "conversion" of extracellular signal to intracellular signal.

The model becomes simple and easy to analyze if you reduce the number of steps. Certain steps can be removed by making certain assumptions such as Equilibrium approximation (EA) or Quasi-Steady-State approximation (QSSA).
If a reversible reaction is extremely fast compared to all other reactions (such as binding-unbinding of the activator to DNA), you can assume that reaction to be in equilibrium and represent the concentration of some species using equilibrium constants.
In Quasi-Steady-State approximation, you assume that certain species are in steady state with respect to others i.e their concentrations do not change over time. In the model you just set $\frac{d[X]}{dt}=0$ for that species. Then obtain the concentration relationships as a function of reaction rate parameters. I'll explain QSSA and EA using the Michaelis-Menten (MM) kinetics.

MM considers following steps:

• Enzyme-Substrate binding (Reversible): $E + S \rightleftharpoons ES$ with forward rate constant k1 and reverse rate constant k2
• Enzyme-Substate complex converting to Product and Enzyme (Irreversible): $ES \to P + E$ with rate constant k3

EA (The actual MM model):
Assuming step 1 is in equilibrium. $k_1[E][S]=k_2[ES]$
Get [ES] in terms of [E] and [S]

QSSA (The revised MM mddel):
Assuming [ES] is in steady state. ${\Large \frac{d[ES]}{dt}}=k_1[E][S]-k_2[ES]-k_3[ES] = 0$
which means $[ES] = \Large \frac{k_1[E][S]}{k_2+k_3}$

This way you have removed certain reactions from the model.

You can do stochastic modeling as well.

For simulations you can use Matlab or Fortran.