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Modeling

Introduction

More and more information on biological networks, and a lot is already known yet. Understanding signaling pathways important for medical health issues, but also for building new networks from scratch using synthetic biology.?
However, the more components a network involves, the harder it gets to estimate how a complex network reacts to changes from both the inside and the outside. This is especially a problem, if entities change both in time and space. Experimentally studying all different components of a network for different possible conditions often is not only time-consumptive but also very expensive and therefore not viable.
Mathematical modeling of biological networks therefore is a powerful tool to support experimental research, predicting the behaviour of networks both concerning time and space. These predictions can then be validated experimentally.

Two crucial biological processes within the DiaCHIP are cell-free expression and the diffusion of proteins to the specific chemical surface. Cell-free expression limits the DiaCHIP regarding time, protein diffusion regarding space thereby determining the maximum protein spot density one can still resolve. Therefore, we decided to model both cell-free expression and protein diffusion to predict the behaviour of protein concentration in time and space.

With a model like this, it is possible to find the minimum time needed to get a sufficiently high concentration of protein for antibody detection. This reduces the time needed for diagnosis using the DiaCHIP. Furthermore, not only protein, but also other molecules' concentrations can be simulated. Therefore, bottlenecks within the biochemical system as well as limiting substrates can be predicted. Specifically increasing the concentration of such limiting substrates as well as substances involved in bottleneck processes may increase not only the overall protein yield but also speed up synthesis to account for an even faster diagnosis.
Moreover, a cell-free expression model has huge potential for other applications.

Overview

With our model we aim to simulate cell-free expression to predict the amount of protein synthesized during the course of time. We set up a system of ordinary differential equations (ODE System) based on the law of mass action, describing the kinetics of the single processes occurring during transcription and translation. Starting from a detailed description including XX different entities and XX different parameters, we defined additional assumptions to simplify our system to a total of ZZ entities and ZZ parameters. Kinetic constants and all other parameter values needed were researched in literature. If no values were found, appropriate values were suggested. As the final model can not be solved analytically, it was solved numerically using Mathematica. [REFERENCE]

Model Decription

1. Transcription

# ODEs: 30 (Simplified: 14) (Shared: cmRNA)
# Parameter: (Simplified: 9) (Shared: lDNA)

Detailed System

[PICTURE] - DONE
[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE] [ODE SYSTEM] - DONE

Simplified System

[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE]
[ODE SYSTEM] - DONE

2. Translation

# ODEs: 61 (Simplified: 52) (Shared: cmRNA)
# Parameter: (Simplified: 32) (Shared: lDNA)

Detailed System

[PICTURE] - DONE
[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE] [ODE SYSTEM] - DONE

Simplified System

[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE] [ODE SYSTEM] - DONE

3. Diffusion

3.1 Introduction and Motivation

In the final step of cellfree expression proteins being produced are diffusing inside the microfluidic chamber. We modeled an ideal case to provide a tool:
On the PDMS slide spots of bound DNA produce proteins with steadily decreasing production rate. The product is distributed homogeneously on the spot and starts diffusing freely in the cell-free mix. Furthermore besides convection through gravitation any interaction is assumed to be negligibly small. The coated iRIf glass is expected to be an ideal sink; any proteins reaching the slide are bound and therefore do not contribute to diffusion anymore.

What knowledge did we want to gain by modeling?
- Time optimization: When is the most efficient time to stop the expression?
- Product optimization: How much of the totally produced proteins does bind to the surface?
- Spot distance optimization: How is the bound protein distributed on the glass slide?

In order to achieve this we constructed the following system.

3.2 Model System

Effective modeling depends on wisely chosen assumptions. Assume the following setup:
(PIC)

Illustrated above is a crosssection of the system around one spot (A1). Due to assumption A2 the number of geometrical degrees of freedom can be reduced to 2.

(PIC: parameters/mats)

Including movement due to gravitation the physical process is described by the diffusion convection equation:

(Formel: Diffgleichung)

It is an inhomogeneous parabolic partial differential equation (PDE). The diffusion constant κ depends on the media and materials involved in the system, the inhomogenity describes sources of the system. The velocity vector v is assumed to be the sedimentation velocity (A4) and therefore a constant. The initial and boundary conditions are the following:

(Formel: ICs and BCs)

The problem can be solved numerically by application of a "finite differences method". The complete space M is split up into squares. Each square inhibits a concentration value. In an iterative method the diffusion between neighbouring squares is calculated in small time steps. If the step size is chosen small enough and the grid fine enough the diffusion can be simulated. The algorithm is the following:

(PIC: Algorithmus)

In a simulation using python (v3.4.2, together with numpy and scipy) depending on time step size, square width, diffusion constant and spot width we could determine binding behaviour on the slide as function of time:

(Eingangsparameter, Graphik oder GIF...)

3.3 Numerical approach

Our approach used the Crank-Nicholson-method: A functions change in one time step is determined by the average of momentary and following time step of the right hand side of the PDE:
(For_07)
In case of convection and diffusion first and second derivations have to be determined. In (finite) differential form these can be approximated by
(For_06)
Proecting those onto our given problem the diffusion-convection equation parts are given by:
(For_10)
where we presupposed Δx=Δz. For the sake of convenience constance had been bundled together
(For_12)
which lead to a large system of equations of the form:
(For_12)
What can be seen is the chosen geometrical approach: Diffusion and convection are done only with direct grid neighbours (two vertical and two horizontal ones). Logically this holds true only for non-border grid points. For the borders boundary conditions have to be considered: We assumed the left and right edges of the box to be open, the upper (PDMS side) and lower side to be closed ones. This can be achieved by the following conditions:
(For_19)

3.4 Assumptions

A1: The spots are equally distributed and distance to the borders of the microfluidic chamber is at least one spot-to-spot distance.

A2: The spots are perfect circles and the binding site at the iRIf glass is homohegeneous -> Around one spot cylindrical symmetry is given.

A3: The area of produced proteins is as thick as the one of bound ones -> χ = ε

A4: Movement by virtue of gravitation is considered to already be in a stationary status (constant movement)