# Team:KULeuven/Modeling

## Introduction

As an introduction to modeling we made a short presentation. This presentation tells about the following:

• some definitions and the role of modeling
• black and white box modeling
• the role of ordinary differential equations
• modeling applied to iGEM

This presentation is mainly based on the presentation of last year and the wiki of ETH Zürich 2007.

## The Full Model

The full model consists of four parts: the blue light receptor, the comparator, the vanillin production and the vanillin receptor, similar to the real world Essencia coli bacteria. The blue light sensor and the vanillin receptor are acting as the inputs of the system. The intensity of blue light determines the wanted vanillin concentration. The vanillin receptor records the actual concentration of vanillin outside the cell. Comparing these two inputs gives us a control system for the vanillin production.

## Modeling steps

The scientific procedure for the study of a physical system can be divided in the following 3 steps

• Parameterization of the system: discover the minimal set of parameters that completely define the system.
• Forward modeling: define the physical laws that, given the values of the parameters of the system, determine the value of the observable parameters.
• Inverse modeling: given observed parameters, infer the actual parameters that produced the observed data.

Most of the time this is an iterative approach, the last step will contain an indication of how good the model fits the observed data. Sometimes it will be necessary to adjust the proposed model.

### Step 1 and 2: Modeling and simulation

By a model, we mean an abstraction of some real system that can be used to obtain predictions and formulate control strategies. In order to be useful, a model must necessarily incorporate elements of two conflicting attributes:

• realism
• simplicity

On the one hand, the model should serve as a reasonably close approximation to the real system, on the other hand, the model must not be so overly complex that it precludes its understanding and manipulation. All that's required is a high correlation between predictions and real-life performance.

We created a single cell model of interacting biochemical reacting to describe the behaviour of our 'Miss Blue Vanilla', using simbiology (Mathworks) graphical representation. The reactions can be described with sufficient accuracy by Ordinary Differential Equations (ODEs) like Mass-Action laws, Hill Kinetic laws and so on. These equations are solved by a deterministic solver available in simbiology.

### Step 3: Inverse modeling

Although some parameters can be obtained by digging in the literature in a search of relevant information, some reactions are still not fully described or exactly known by science. This requires another method for obtaining the needed information: inverse modeling also known as parameter estimation.

While the first two steps are mainly deductive, this step is inductive. The inverse problem consists of using the actual result of some measurements to infer the value of the parameters that characterizes the system. While the forward problem has (in deterministic physics) a unique solution, the inverse problem does not. The most general theory is obtained when using a probabilistic point of view, where the a priori information on the model parameters is represented by a probability distribution over the 'model space'. Because an exhaustive search through the model space is computationally very demanding, more intelligent Monte Carlo techniques will be used. We used a Metropolis-Hastings sampling algorithm. Notice that if one wants to resolve all the parameters in the model, a vast number of experimental data has to be obtained.

Most of the time we are only interested in the model that fits the observed data best. This best model can be obtained by the solution of a (non linear) optimization problem. This is the main method we will use to estimate the unknown parameters in the model. Although the probabilistic approach is more general it suffers from major computational overhead.