Difference between revisions of "Team:Freiburg/Results/Modeling"

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<h2>3. Diffusion<\h2>
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<h3>3.1 Introduction and Motivation<\h3>
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In the final step of cellfree expression proteins produced are diffunding inside the microfluidic camber. We observed
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at the ideal case:
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<br>
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On the PDMS slide circle spots of bound DNA produce proteins with steadily decreasing production
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rate. The product is distributed homogeneously on the spot and starts diffunding freely in the cellfree
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mix without considering any interactions. The coated iRIf glass is an ideal sink; any proteins reaching the
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the slide are bound and therefore do not contribute to diffusion anymore.
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<br><br>
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What knowledge did we want to gain by modeling?
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<br>
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- Time optimization: When is the most efficient time to stop the expression?
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<br>
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- Product optimization: How much of the totally produced proteins does bind to the surface?
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<br>
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- Spot distance optimization: How is the bound protein distributed on the glass slide?
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<br><br>
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In order to achiev this we constructed the following system.
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<br><br>
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<h3>3.2 Model System<\h3>
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Effective modelling depends on a wisely chosen assumptions
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Assume the following setup:
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<br>
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(PIC)
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<br>
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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.
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<br><br>
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(PIC: parameters)
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<br><br>
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The physical process is described by the diffusion equation:
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<br><br>
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(Formel: Diffgleichung)
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<br><br>
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It is an inhomogeneous parabolic partial differential equation (PDE). The diffusion constant &kappa; depends on the media and materials involved in the system, the inhomogenity describes sources and sinks inside the system. The initial and boundary conditions are the following:
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<br><br>
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(Formel: ICs and BCs)
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<br><br>
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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:
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<br><br>
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(PIC: Algorithmus)
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<br><br>
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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:
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<br><br>
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(Eingangsparameter, Graphik oder GIF...)
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<br><br>
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<h3>3.3 Assumptions<\h3>
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A1: The spots are equally distributed and distance to the borders of the microfluidic chamber is at least one
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spot-to-spot distance.
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<br><br>
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A2: The spots are perfect circles and the binding site at the iRIf glass is homohegeneous -> Around one spot
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cylindrical symmetry is given.
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<br><br>
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A3: The area of produced proteins is as thick as the one of bound ones -> &chi; = &epsilon;
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<br><br>
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A4: ...
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<br><br>
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Revision as of 17:59, 13 September 2015

""

Modeling

Note

Introduction

Stilbruch hier. Das ist eine resultspage introduction nd results waren bisher immer getrennt. (Stefan) 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 whole 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 DIA Chip are cell-free expression and the diffusion of proteins to the chemical surface. Cell-free expression limits the DIA chip regarding time, protein diffusion regarding space, determinig 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 DIA Chip. 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. Hier fehlen noch Beispiele.

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]

Detailed System

Transcription

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

ODE System


\[ tc0(1): \;\;\; \frac{dc^{RNAP}_{free}[t]}{dt}\: =\: c^{RNAP}_{bound}[t] \cdot k^{RNAP}_{gain} - c^{RNAP}_{free}[t] \cdot k^{RNAP}_{loss} \] \[ \hphantom{tc0(1): \;\;\; \frac{dc^{RNAP}_{free}[t]}{dt}\: =\: } + c^{RNAP}_{sigma}[t] \cdot k^{sigma}_{off} - c^{RNAP}_{free}[t] \cdot c^{sigma}[t] \cdot k^{sigma}_{on} \] \[ \hphantom{tc0(1): \;\;\; \frac{dc^{RNAP}_{free}[t]}{dt}\: =\: } + c^{RNAP}_{elongter}[-1][t] \cdot k^{RNAP}_{diss} \]
\[ tc0(2): \;\;\; \frac{dc^{sigma}[t]}{dt}\: =\: c^{sigma}_{bound}[t] \cdot k^{sigma}_{gain} - c^{sigma}[t] \cdot k^{sigma}_{loss} + c^{RNAPsigma}_{bound}[t] \cdot k^{sigma}_{off} \] \[ \hphantom{tc0(2): \;\;\; \frac{dc^{sigma}[t]}{dt}\: =\: } + c^{RNAP}_{sigma}[t] \cdot k^{sigma}_{off} - c^{RNAP}_{free}[t] \cdot c^{sigma}[t] \cdot k^{sigma}_{on} \] \[ \hphantom{tc0(2): \;\;\; \frac{dc^{sigma}[t]}{dt}\: =\: } + c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{prel} \]
\[ tc0(3.1): \;\;\; \frac{dc^{RNAP}_{sigmaint}[t]}{dt}\: =\: c^{RNAP}_{free}[t] \cdot c^{sigma}[t] \cdot k^{sigma}_{on} - c^{RNAP}_{sigmaint}[t] \cdot k^{sigma}_{off} \] \[ \hphantom{tc0(3.1): \;\;\; \frac{dc^{RNAP}_{sigmaint}[t]}{dt}\: =\: } + c^{RNAP}_{sigma}[t] \cdot k^{RNAPsigma}_{isore} - c^{RNAP}_{sigmaint}[t] \cdot k^{RNAPsigma}_{iso} \] \[ tc0(3.2): \;\;\; \frac{dc^{RNAP}_{sigma}[t]}{dt}\: =\: c^{RNAPsigma}_{bound}[t] \cdot k^{RNAP}_{gain} \cdot c^{RNAP}_{sigma}[t] \cdot k^{RNAP}_{loss} \] \[ \hphantom{tc0(3.2): \;\;\; \frac{dc^{RNAP}_{sigma}[t]}{dt}\: =\: } + \sum \limits_{i=0}^n c^{RNAP}_{on}[i][t] \cdot k^{RNAP}_{off} - c^{RNAP}_{sigma}[t] \cdot p^{DNA} \cdot l^{DNA} \cdot k^{RNAP}_{on} \] \[ \hphantom {tc0(3.2): \;\;\; \frac{dc^{RNAP}_{sigma}[t]}{dt}\: =\: } + c^{RNAP}_{sigmaint}[t] \cdot k^{sigma}_{iso} - c^{RNAP}_{sigma}[t] \cdot k^{RNAPsigma}_{isore} \]
\[ tc0(4): \;\;\; \frac{dc^{RNAP}_{on}[i][t]}{dt}\: =\: c^{RNAP}_{sigma}[t] \cdot p^{DNA} \cdot k^{RNAP}_{on} \] \[ \hphantom{tc0(4): \;\;\; \frac{dc^{RNAP}_{on}[i][t]}{dt}\: =\: } + c^{RNAP}_{on}[i\:-\:v^{RNAP}_{move} \cdot dt][t] \cdot (1 - k^{RNAP}_{off}) - c^{RNAP}_{on}[i][t] \]
\[ tc0(5): \;\;\; \frac{dc^{RNAP}_{prom}[t]}{dt}\: =\: \sum \limits_{i\:=\:n - v^{RNAP}_{move} \cdot dt}^n c^{RNAP}_{on}[i][t] \cdot (1 - k^{RNAP}_{off}) \] \[ \hphantom{tc0(5): \;\;\; \frac{dc^{RNAP}_{prom}[t]}{dt}\: =\: } + c^{RNAP}_{open}[t] \cdot k^{tc}_{closed} - c^{RNAP}_{prom}[t] \cdot k^{tc}_{open} \]
\[ tc0(6): \;\;\; \frac{dc^{RNAP}_{open}[t]}{dt}\: =\: c^{RNAP}_{prom}[t] \cdot k^{tc}_{open} - c^{RNAP}_{open}[t] \cdot k^{tc}_{closed} \] \[ \hphantom{tc0(6): \;\;\; \frac{dc^{RNAP}_{open}[t]}{dt}\: =\: } + c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{iniab} - c^{RNAP}_{open}[t] \cdot c^{ATP}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{ini1} \]
\[ tc0(7): \;\;\; \frac{dc^{RNAP}_{ini1}[t]}{dt}\: =\: c^{RNAP}_{open}[t] \cdot c^{ATP}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{ini1} - c^{RNAP}_{ini1}[t] \cdot c^{X_2 TP}[t] \cdot k^{tc}_{inix} \]
\[ tc0(8.1): \;\;\; \frac{dc^{RNAP}_{ini}[i][t]}{dt}\: =\: c^{RNAP}_{ini}[i-1][t] \cdot c^{X_i TP}[t] \cdot k^{tc}_{inix} - c^{RNAP}_{ini}[i][t] \cdot c^{X_i+1 TP}[t] \cdot k^{tc}_{inix}, \] \[ \hphantom{tc0(8.1): \;\;\; \frac{dc^{RNAP}_{ini}[i][t]}{dt}\: =\: } (i = 2, ..., l^{ini-1}) \] \[ tc0(8.2): \;\;\; \frac{dc^{RNAP}_{ini}[1][t]}{dt}\: =\: \frac{dc^{RNAP}_{ini1}[t]}{dt} \] \[ tc0(8.3): \;\;\; \frac{dc^{RNAP}_{ini}[-1][t]}{dt}\: =\: c^{RNAP}_{ini}[-2][t] \cdot c^{X_-1 TP}[t] \cdot k^{tc}_{inix} - c^{RNAP}_{ini}[-1][t] \cdot (k^{tc}_{iniab} + k^{tc}_{prel}) \]
\[ tc0(9): \;\;\; \frac{dc^{RNAP}_{prel}[t]}{dt}\: =\: c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{prel} - c^{RNAP}_{prel}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} \]
\[ tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: c^{RNAP}_{elong}[i-1][t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_i TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: } - c^{RNAP}_{elong}[i][t] \cdot ((1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} + prob^{tc}_{mm} \cdot (c^{NTPs}[t] - c^{X_1 TP}[t]) \cdot k^{tc}_{elong}) \] \[ \hphantom{tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: } + c^{RNAP}_{elongGreAB}[j + l^{mRNA}_{cl}][t] \cdot k^{GreAB}_{cat}, \] \[ \hphantom{tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: } (i = 2, ..., l^{elong-1}),\: (j = i\: and\: j = 2, ..., l^{elong} - l^{mRNA}_{cl}) \] \[ tc0(10.2): \;\;\; \frac{dc^{RNAP}_{elong}[1][t]}{dt}\: =\: c^{RNAP}_{prel}[t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(10.2): \;\;\; \frac{dc^{RNAP}_{elong}[1][t]}{dt}\: =\: } - c^{RNAP}_{elong}[1][t] \cdot ((1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} + prob^{tc}_{mm} \cdot (c^{NTPs}[t] - c^{X_1 TP}[t]) \cdot k^{tc}_{elong}) \] \[ \hphantom{tc0(10.2): \;\;\; \frac{dc^{RNAP}_{elong}[1][t]}{dt}\: =\: } + c^{RNAP}_{elongGreAB}[l^{mRNA}_{cl}][t] \cdot k^{GreAB}_{cat} \] \[ tc0(10.3): \;\;\; \frac{dc^{RNAP}_{elong}[-1][t]}{dt}\: =\: c^{RNAP}_{elong}[-2][t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_{-1} TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(10.3): \;\;\; \frac{dc^{RNAP}_{elong}[-1][t]}{dt}\: =\: } - c^{RNAP}_{elong}[-1][t] \cdot l^{mRNA} \cdot c^{pprot} * k^{pprot}_{on} \]
\[ tc0(11.1): \;\;\; \frac{dc^{RNAP}_{elongter}[i][t]}{dt}\: =\: (c^{RNAP}_{elongter}[i-1][t] - c^{RNAP}_{elongter}[i][t]) \cdot c^{ATP}[t] \cdot k^{pprot}_{cat} \] \[ \hphantom{tc0(11.1): \;\;\; \frac{dc^{RNAP}_{elongter}[i][t]}{dt}\: =\: } + c^{RNAP}_{elong}[i] \cdot c^{pprot} \cdot k^{pprot}_{on}, \] \[ \hphantom{tc0(11.1): \;\;\; \frac{dc^{RNAP}_{elongter}[i][t]}{dt}\: =\: } (i = 2, ..., l^{mRNA}-1) \] \[ tc0(11.2): \;\;\; \frac{dc^{RNAP}_{elongter}[-1][t]}{dt}\: =\: c^{RNAP}_{elongter}[-2][t] \cdot c^{ATP}[t] \cdot k^{pprot}_{cat} - c^{RNAP}_{elongter}[-1][t] * k^{RNAP}_{diss} \] \[ \hphantom{tc0(11.2): \;\;\; \frac{dc^{RNAP}_{elongter}[-1][t]}{dt}\: =\: } + c^{RNAP}_{elong}[-1][t] \cdot c^{pprot}[t] \cdot k^{pprot}_{on} \]
\[ tc0(12): \;\;\; \frac{dc^{mRNA}[t]}{dt}\: =\: c^{RNAP}_{elongter}[-1][t] \cdot k^{RNAP}_{diss} - c^{RNAse}_{onmRNA}[t] \cdot k^{RNAse}_{cat} \]
\[ tc0(13): \;\;\; \frac{dc^{RNAP}_{elongmm}[i][t]}{dt}\: =\: c^{RNAP}_{elong}[i-1][t] \cdot prob^{tc}_{mm} \cdot (c^{NTPs}[t] - c^{X_1 TP}[t]) \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(13): \;\;\; \frac{dc^{RNAP}_{elongmm}[i][t]}{dt}\: =\: } - c^{RNAP}_{elongmm}[i][t] \cdot c^{GreAB}[t] \cdot k^{GreAB}_{on} \]
\[ tc0(14): \;\;\; \frac{dc^{RNAP}_{elongGreAB}[i][t]}{dt}\: =\: c^{RNAP}_{elongmm}[i][t] \cdot c^{GreAB}[t] \cdot k^{GreAB}_{on} - c^{RNAP}_{elongGreAB}[i][t] \cdot k^{GreAB}_{cat} \]
\[ tc0(15): \;\;\; \frac{dc^{RNAse}[t]}{dt}\: =\: c^{RNAse}_{bound}[t] \cdot k^{RNAse}_{gain} - c^{RNAse}[t] \cdot k^{RNAse}_{loss} \] \[ \hphantom{tc0(15): \;\;\; \frac{dc^{RNAse}[t]}{dt}\: =\: } + (c^{RNAse}_{onmRNAcl}[t] + c^{RNAse}_{onmRNAab}[t] + c^{RNAse}_{onmRNA}[t]) \cdot k^{RNAse}_{cat} \] \[ \hphantom{tc0(15): \;\;\; \frac{dc^{RNAse}[t]}{dt}\: =\: } - (c^{mRNAcl}[t] + c^{mRNAab}[t] + c^{mRNA}[t]) \cdot c^{RNAse}[t] \cdot k^{RNAse}_{on} \]
\[ tc0(16): \;\;\; \left( \frac{dc^{RNAse}_{onmRNA}[t]}{dt} ,\: \frac{dc^{RNAse}_{onmRNAab}[t]}{dt} ,\: \frac{dc^{RNAse}_{onmRNAcl}[t]}{dt} \right) \] \[ \hphantom{tc0(16): \;\;\; } = c^{RNAse}[t] \cdot (c^{mRNA}[t],\: c^{mRNAab}[t],\: c^{mRNAcl}[t]) \cdot k^{RNAse}_{on} - \left( c^{RNAse}_{onmRNA}[t],\: c^{RNAse}_{onmRNAab}[t],\: c^{RNAse}_{onmRNAcl}[t] \right) \cdot k^{RNAse}_{cat} \]
\[ tc0(17): \;\;\; \frac{dc^{mRNAab}[t]}{dt}\: =\: c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{iniab} - c^{RNAse}_{onmRNAab}[t] \cdot k^{RNAse}_{cat} \]
\[ tc0(18): \;\;\; \frac{dc^{mRNAcl}[t]}{dt}\: =\: \sum \limits_{i=1}^n c^{RNAP}_{elongGreAB}[i][t] \cdot k^{GreAB}_{cat} + 2 \cdot c^{RNAse}_{onmRNA}[t] \cdot k^{RNAse}_{cat} - c^{RNAse}_{onmRNAcl}[t] \cdot k^{RNAse}_{cat} \]
\[ tc0(19.1): \;\;\; \frac{dc^{entity}_{bound}[t]}{dt}\: =\: c^{entity}[t] \cdot k^{entity}_{loss} - c^{entity}_{bound}[t] \cdot k^{entity}_{gain}, \] \[ \hphantom{tc0(19.1): \;\;\; \frac{dc^{entity}_{bound}[t]}{dt}\: =\: } (entity \notin \{RNAP, RNAPsigma\}) \]
\[ tc0(19.2): \;\;\; \left( \frac{dc^{RNAP}_{bound}[t]}{dt},\: \frac{dc^{RNAPsigma}_{bound}[t]}{dt} \right) \] \[ \hphantom{tc0(19.2): \;\;\; } = c^{RNAP}[t] \cdot k^{RNAP}_{loss} - c^{RNAP}_{bound}[t] \cdot k^{RNAP}_{gain} + \left( c^{RNAPsigma}_{bound}[t],\: - c^{RNAPsigma}_{bound}[t] \right) \cdot k^{sigma}_{off} \]
\[ tc0(20): \;\;\; \frac{dc^{pprot}[t]}{dt}\: =\: c^{pprot}_{bound}[t] \cdot k^{pprot}_{gain} - c^{pprot}[t] \cdot k^{pprot}_{loss} \] \[ \hphantom{tc0(20): \;\;\; \frac{dc^{pprot}[t]}{dt}\: =\: } + c^{RNAP}_{elongter}[-1][t] \cdot k^{RNAP}_{diss} - c^{RNAP}_{elong}[-1][t] \cdot l^{mRNA} \cdot c^{pprot}[t] \cdot k^{pprot}_{on} \]
\[ tc0(21): \;\;\; \frac{dc^{GreAB}[t]}{dt}\: =\: c^{GreAB}_{bound}[t] \cdot k^{GreAB}_{gain} - c^{GreAB}[t] \cdot k^{GreAB}_{loss} \] \[ \hphantom{tc0(21): \;\;\; \frac{dc^{GreAB}[t]}{dt}\: =\: } + \sum \limits_{i=1}^n c^{RNAP}_{elongGreAB}[i][t] \cdot k^{GreAB}_{cat} - \sum \limits_{i=1}^n c^{RNAP}_{elongmm}[i][t] \cdot c^{GreAB}[t] \cdot k^{GreAB}_{on} \]
\[ tc0(22): \;\;\; \frac{dc^{NTP}[t]}{dt}\: =\: - \sum \limits_{i=2, X_i=N}^n c^{RNAP}_{ini}[i-1][t] \cdot c^{X_i TP}[t] \cdot k^{tc}_{inix} \] \[ \hphantom{tc0(22): \;\;\; \frac{dc^{NTP}[t]}{dt}\: =\: } - c^{RNAP}_{prel}[t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(22): \;\;\; \frac{dc^{NTP}[t]}{dt}\: =\: } - \sum \limits_{i=2, X_i=N}^{n-1} c^{RNAP}_{elong}[i-1][t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_i TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(22): \;\;\; \frac{dc^{NTP}[t]}{dt}\: =\: } \left[ - c^{RNAP}_{open}[t] \cdot c^{ATP}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{ini1}\right]_{for X_1 = N} \] \[ \hphantom{tc0(22): \;\;\; \frac{dc^{NTP}[t]}{dt}\: =\: } \left[ - c^{RNAP}_{open}[t] \cdot c^{ATP}[t] \cdot c^{NTP}[t] \cdot k^{tc}_{ini1} - \sum \limits_{i=1}^{n-1} c^{RNAP}_{elongter}[i][t] \cdot c^{ATP}[t] \cdot k^{pprot}_{cat}\right]_{for N = A} \]
\[ tc0(23): \;\;\; \frac{dc^{NTPs}[t]}{dt}\: =\: \frac{dc^{ATP}[t]}{dt} + \frac{dc^{TTP}[t]}{dt} + \frac{dc^{GTP}[t]}{dt} + \frac{dc^{CTP}[t]}{dt} \]

Translation

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

ODE System


Amino Acid Activation


\[ tl0(1): \;\;\; \frac{dc^{AaxtRNASyn}[t]}{dt}\: =\: c^{AaxtRNASyn}_{AaatRNA}[t] \cdot k^{AaxtRNASyn}_{cat2} \] \[ \hphantom{tl0(1): \;\;\; \frac{dc^{AaxtRNASyn}[t]}{dt}\: =\: } + c^{AaxtRNASyn}_{Aax}[t] \cdot k^{Aax}_{off} - c^{AaxtRNASyn}_{Aax}[t] \cdot c^{Aax}[t] \cdot k^{Aax}_{on} \]
\[ tl0(2): \;\;\; \frac{dc^{AaxtRNASyn}_{Aax}[t]}{dt}\: =\: c^{AaxtRNASyn}[t] \cdot c^{Aax}[t] \cdot k^{Aax}_{on} - c^{AaxtRNASyn}_{Aax}[t] \cdot k^{Aax}_{off} \] \[ \hphantom{tl0(2): \;\;\; \frac{dc^{AaxtRNASyn}_{Aax}[t]}{dt}\: =\: } - c^{AaxtRNASyn}_{Aax}[t] \cdot c^{ATP}[t] \cdot k^{AaxtRNASyn}_{cat1} \]
\[ tl0(3): \;\;\; \frac{dc^{AaxtRNASyn}_{Aaa}[t]}{dt}\: =\: c^{AaxtRNASyn}_{Aax}[t] \cdot c^{ATP}[t] \cdot k^{AaxtRNASyn}_{cat1} \] \[ \hphantom{tl0(3): \;\;\; \frac{dc^{AaxtRNASyn}_{Aaa}[t]}{dt}\: =\: } + c^{AaxtRNASyn}_{AaatRNA}[t] \cdot k^{tRNA}_{off} - c^{AaxtRNASyn}_{AaatRNA}[t] \cdot k^{AaxtRNASyn}_{cat2} \]
\[ tl0(4): \;\;\; \frac{dc^{AaxtRNASyn}_{AaatRNA}[t]}{dt}\: =\: + c^{AaxtRNASyn}_{Aaa}[t] \cdot c^{tRNA}[t] \cdot k^{tRNA}_{on} - c^{AaxtRNASyn}_{AaatRNA}[t] \cdot k^{tRNA}_{off} \] \[ \hphantom{tl0(4): \;\;\; \frac{dc^{AaxtRNASyn}_{AaatRNA}[t]}{dt}\: =\: } - c^{AaxtRNASyn}_{AaatRNA}[t] \cdot k^{AaxtRNASyn}_{cat2} \]
\[ tl0(5): \;\;\; \frac{dc^{AaxtRNA}[t]}{dt}\: =\: c^{AaxtRNASyn}_{AaatRNA}[t] \cdot k^{AaxtRNASyn}_{cat2} \] \[ \hphantom{tl0(5): \;\;\; \frac{dc^{AaxtRNA}[t]}{dt}\: =\: } + c^{AaxtRNA}_{EFTu}[t] \cdot k^{EFTu}_{off} - c^{AaxtRNA}[t] \cdot c^{EFTu}[t] \cdot k^{EFTu}_{on} \] \[ \hphantom{tl0(5): \;\;\; \frac{dc^{AaxtRNA}[t]}{dt}\: =\: } + c^{tl}_{el1mm}[t] \cdot k^{AaxtRNA}_{reject} \]
\[ tl0(6): \;\;\; \frac{dc^{ftrans}[t]}{dt}\: =\: c^{ftrans}_{MettRNAfMet}[t] \cdot k^{MettRNAfMet}_{off} - c^{ftrans}[t] \cdot c^{MettRNAfMet}[t] \cdot k^{MettRNAfMet}_{on} \] \[ \hphantom{tl0(6): \;\;\; \frac{dc^{ftrans}[t]}{dt}\: =\: } + c^{ftrans}_{MettRNAfMet}[t] \cdot c^{f}[t] \cdot k^{ftrans}_{cat} - c^{ftrans}[t] \cdot c^{fMettRNAfMet}[t] \cdot c^{t}[t] \cdot k^{ftrans}_{deg} \]
\[ tl0(7): \;\;\; \frac{dc^{ftrans}_{MettRNAfMet}[t]}{dt}\: =\: c^{ftrans}[t] \cdot c^{MettRNAfMet}[t] \cdot k^{MettRNAfMet}_{on} - c^{ftrans}_{MettRNAfMet}[t] \cdot k^{MettRNAfMet}_{off} \] \[ \hphantom{tl0(7): \;\;\; \frac{dc^{ftrans}_{MettRNAfMet}[t]}{dt}\: =\: } + c^{ftrans}[t] \cdot c^{fMettRNAfMet}[t] \dot c^{t}[t] \cdot k^{ftrans}_{deg} - c^{ftrans}_{MettRNAfMet}[t] \cdot c^{f}[t] \cdot k^{ftrans}_{cat} \]
\[ tl0(8): \;\;\; \frac{dc^{fMettRNAfMet}[t]}{dt}\: =\: c^{ftrans}_{MettRNAfMet}[t] \cdot c^{f}[t] \cdot k^{ftrans}_{cat} - c^{ftrans}[t] \cdot c^{fMettRNAfMet}[t] \cdot c^{t}[t] \cdot k^{ftrans}_{deg} \] \[ \hphantom{tl0(8): \;\;\; \frac{dc^{fMettRNAfMet}[t]}{dt}\: =\: } + c^{tl}_{ini2}[t] \cdot k^{fMettRNAfMet}_{off} - c^{tl}_{ini1}[t] \cdot c^{fMettRNAfMet}[t] \cdot k^{fMettRNAfMet}_{on} \]

Initialization


\[ tl0(9): \;\;\; \frac{dc^{70s}[t]}{dt}\: =\: c^{30s}[t] \cdot c^{50s}[t] \cdot k^{50s}_{bind} - c^{70s}[t] \cdot c^{IF3}[t] \cdot k^{IF3}_{on} \]
\[ tl0(10): \;\;\; \frac{dc^{70sIF3}[t]}{dt}\: =\: c^{70s}[t] \cdot c^{IF3}[t] \cdot k^{IF3}_{on} - c^{70sIF3}[t] \cdot k^{IF3}_{off} \] \[ \hphantom{tl0(10): \;\;\; \frac{dc^{70sIF3}[t]}{dt}\: =\: } - c^{70sIF3}[t] \cdot k^{50s}_{diss} \]
\[ tl0(11): \;\;\; \frac{dc^{30sIF3}[t]}{dt}\: =\: c^{70sIF3}[t] \cdot k^{50s}_{diss} - c^{30sIF3}[t] \cdot k^{IF3}_{off} \] \[ \hphantom{tl0(11): \;\;\; \frac{dc^{30sIF3}[t]}{dt}\: =\: } + c^{30sIF3IF1}[t] \cdot k^{IF1}_{off} - c^{30sIF3}[t] \cdot c^{IF1}[t] \cdot k^{IF1}_{on} \]
\[ tl0(12): \;\;\; \frac{dc^{30sIF3IF1}[t]}{dt}\: =\: c^{30sIF3}[t] \cdot c^{IF1}[t] \cdot k^{IF1}_{on} - c^{30sIF3IF1}[t] \cdot k^{IF1}_{off} \] \[ \hphantom{tl0(12): \;\;\; \frac{dc^{30sIF3IF1}[t]}{dt}\: =\: } + c^{tl}_{ini0}[t] \cdot k^{mRNA}_{off} - c^{30sIF3IF1}[t] \cdot c^{mRNA}[t] \cdot k^{mRNA}_{on} \]
\[ tl0(13): \;\;\; \frac{dc^{tl}_{ini0}[t]}{dt}\: =\: c^{30sIF3IF1}[t] \cdot c^{mRNA}[t] \cdot k^{mRNA}_{on} - c^{tl}_{ini0}[t] \cdot k^{mRNA}_{off} \] \[ \hphantom{tl0(13): \;\;\; \frac{dc^{tl}_{ini0}[t]}{dt}\: =\: } + c^{tl}_{ini1}[t] \cdot k^{IF2}_{off} - c^{tl}_{ini0}[t] \cdot c^{IF2}[t] \cdot k^{IF2}_{on} \]
\[ tl0(14): \;\;\; \frac{dc^{tl}_{ini1}[t]}{dt}\: =\: c^tl_ini0[t] * c^IF2[t] * k^IF2_on - c^tl_ini1[t] * k^IF2_off + c^tl_ini2[t] * k^fMettRNAfMet_off - c^tl_ini1[t] * c^fMettRNAfMet[t] * k^fMettRNAfMet_on tl0(15): dc^tl_ini2[t]/dt = c^tl_ini1[t] * c^fMettRNAfMet[t] * k^fMettRNAfMet_on - c^tl_ini2[t] * k^fMettRNAfMet_off c^tl_inidissIF3[t] * c^IF3[t] * k^IF3_bind - c^tl_ini2[t] * k^IF3_diss tl0(16): dc^tl_inidissIF3[t]/dt = c^tl_ini2[t] * k^IF3_diss - c^tl_inidissIF3[t] * c^IF3[t] * k^IF3_bind + c^tl_inion50s[t] * k^50s_off - c^tl_inidissIF3[t] * c^50s[t] * k^50s_on tl0(17): dc^tl_inion50s[t]/dt = c^tl_inidissIF3[t] * c^50s[t] * k^50s_on - c^tl_inion50s[t] * k^50s_off + c^tl_inidissIF1[t] * c^IF1[t] * k^IF1_bind - c^tl_inion50s[t] * k^IF1_diss tl0(18): dc^tl_inidissIF1[t]/dt = c^tl_inion50s[t] * k^IF1_diss - c^tl_inidissIF1[t] * c^IF1[t] * k^IF1_bind - c^tl_inidissIF1[t] * k^IF2_cat tl0(19): dc^tl_inicat[t]/dt = c^tl_inidissIF1[t] * k^IF2_cat - c^tl_inicat[t] * k^IF2_diss tl0(20): dc^IF1[t]/dt = c^30sIF3IF1[t] * k^IF1_off - c^30sIF3[t] * c^IF1[t] * k^IF1_on + c^tl_inion50s[t] * k^IF1_diss - c^tl_inidissIF1[t] * c^IF1[t] * k^IF1_bind tl0(21): dc^IF3[t]/dt = (c^70sIF3[t] + c^30sIF3[t]) * k^IF3_off - (c^70s + c^30s) * k^IF3_on + c^tl_ini2[t] * k^IF3_diss - c^tl_inidissIF3[t] * c^IF3[t] * k^IF3_bind tl0(22): dc^IF2[t]/dt = c^tl_ini1[t] * k^IF2_off - c^tl_ini0[t] * c^IF2[t] * k^IF2_on + c^IF2_GDP[t] * c^GTP[t] * k^IF2_reg - c^IF2[t] * c^GDP[t] * k^IF2_deg tl0(23): dc^IF2_GDP[t]/dt = c^tl_inicat[t] * k^IF2_diss + c^IF2[t] * c^GDP[t] * k^IF2_deg - c^IF2_GDP[t] * c^GTP[t] * k^IF2_reg

Elongation.


tl0(24.1): dc^tl_el0[i][t]/dt = c^tl_el0AaxtRNA[i][t] * k^AaxtRNAEFTu_off - c^tl_el0[i] * c^AaxtRNA_EFTu[t] * k^AaxtRNAEFTu_on + c^tl_el1mm[i][t] * k^tRNA_reject + c^tl_el3[i-1] * k^tRNA_dissel, (i = 2, ..., l^mRNAcodons) tl0(24.2): dc^tl_el0[1][t]/dt = c^tl_inicat[t] * k^IF2_diss + c^tl_el0AaxtRNA[0][t] * k^AaxtRNAEFTu_off - c^tl_el0[0] * c^AaxtRNA_EFTu[t] * k^AaxtRNAEFTu_on + c^tl_el1mm[0][t] * k^tRNA_reject tl0(25): dc^tl_el0AaxtRNA[i][t]/dt = c^tl_el0[i] * c^AaxtRNA_EFTu[t] * k^AaxtRNAEFTu_on - c^tl_el0AaxtRNA[i][t] * k^AaxtRNAEFTu_off - c^tl_el0AaxtRNA[i][t] * k^EFTu_cat, (i = 1, ..., l^mRNAcodons) tl0(26): dc^tl_el0cat[i][t]/dt = c^tl_el0AaxtRNA[i][t] * k^EFTu_cat - c^tl_el0cat[i][t] * k^EFTu_diss, (i = 1, ..., l^mRNAcodons) tl0(27): dc^tl_el1raw[i][t]/dt = c^tl_el0cat[i][t] * k^EFTu_diss - c^tl_el1raw[i][t] * k^tl_proof, (i = 1, ..., l^mRNAcodons) tl0(28): dc^tl_el1proof[i][t]/dt = c^tl_el1raw[i][t] * k^tl_proof - c^tl_el1proof[i][t] * (prob^tl_mm + (1 - prob^tl_mm)), (i = 1, ..., l^mRNAcodons) tl0(29): dc^tl_el1correct[i][t]/dt = c^tl_el1proof[i][t] * (1 - prob^tl_mm) - c^tl_el1correct[i][t] * k^tl_elong, (i = 1, ..., l^mRNAcodons) tl0(30): dc^tl_el2[i][t]/dt = c^tl_el1correct[i][t] * k^tl_elong + c^tl_el2EFG[i][t] * k^EFG_off - c^tl_el2[i][t] * c^EFG[t] * k^EFG_on, (i = 1, ..., l^mRNAcodons) tl0(31.1): dc^tl_el2EFG[i][t]/dt = c^tl_el2[i][t] * c^EFG[t] * k^EFG_on - c^tl_el2EFG[i][t] * k^EFG_off - c^tl_el2EFG[i][t] * k^EFG_cat, (i = 1, ..., l^mRNAcodons) tl0(31.2): dc^tl_el2EFGcat[i][t]/dt = c^tl_el2EFG[i][t] * k^EFG_cat - c^tl_el2EFGcat[i][t] * k^tl_trans, (i = 1, ..., l^mRNAcodons) tl0(31.3): dc^tl_el2EFGtrans[i][t]/dt = c^tl_el2EFGcat[i][t] * k^tl_trans - c^tl_el2EFGtrans[i][t] * k^EFG_diss (i = 1, ..., l^mRNAcodons) tl0(32): dc^tl_el3[i][t]/dt = c^tl_el2EFGtrans[i][t] * k^EFG_diss - c^tl_el3[i][t] * k^tRNA_dissel (i = 1, ..., l^mRNAcodons) tl0(33): dc^AaxtRNA_EFTu[t]/dt = c^AaxtRNA[t] * c^EFTu[t] * k^EFTu_on - c^AaxtRNA_EFTu[t] * k^EFTu_off + \sum \limits_{i=1}^n c^tl_el0AaxtRNA[i][t] * k^AaxtRNAEFTu_off - \sum \limits_{i=1}^n c^tl_el0[i][t] * c^AaxtRNA_EFTu[t] * k^AaxtRNAEFTu_on tl0(34): dc^tl_el1mm[i][t]/dt = c^tl_el1raw[i][t] * prob^tl_mm - c^tl_el1mm[i][t] * k^AaxtRNA_reject (i = 1, ..., l^mRNAcodons) tl0(35): dc^EFTu_GDP[t]/dt = \sum \limits_{i=1}^n c^tl_el0cat[i][t] * k^EFTu_diss + c^EFTu_GDPEFTS[t] * k^EFTS_off - c^EFTu_GDP[t] * c^EFTS[t] * k^EFTS_on tl0(36): dc^EFTu_GDPEFTS[t]/dt = c^EFTu_GDP[t] * c^EFTS[t] * k^EFTS_on - c^EFTu_GDPEFTS[t] * k^EFTS_off + c^EFTu_EFTS[t] * c^GDP[t] * k^GDP_bindTu - c^EFTu_GDPEFTS[t] * k^GDP_dissTu tl0(37): dc^EFTu_EFTS[t]/dt = c^EFTu_GDPEFTS[t] * k^GDP_dissTu - c^EFTu_EFTS[t] * c^GDP[t] * k^GDP_bindTu + c^EFTu_GTPEFTS[t] * k^GTP_offTu - c^EFTu_EFTS[t] * c^GTP[t] * k^GTP_onTu tl0(38): dc^EFTu_GTPEFTS[t]/dt = c^EFTu_EFTS[t] * c^GTP[t] * k^GTP_onTu - c^EFTu_GTPEFTS[t] * k^GTP_offTu + c^EFTu[t] * c^EFTS[t] * k^EFTS_bind - c^EFTu_GTPEFTS[t] * k^EFTS_diss tl0(39): dc^EFTu[t]/dt = c^AaxtRNA_EFTu[t] * k^EFTu_off - c^EFTu[t] * c^AaxtRNA[t] * k^EFTu_on + c^EFTu_GTPEFTS[t] * k^EFTS_diss - c^EFTu[t] * c^EFTS[t] * k^EFTS_bind tl0(40): dc^EFTS[t]/dt = c^EFTu_GDPEFTS[t] * k^EFTS_off - c^EFTu_GDP[t] * c^EFTS[t] * k^EFTS_on + c^EFTu_GTPEFTS[t] * k^EFTS_diss - c^EFTu[t] * c^EFTS[t] * k^EFTS_bind tl0(41): dc^EFG_GDP[t]/dt = \sum \limits_{i=1}^n c^tl_el2EFGtrans[i][t] * k^EFG_diss + c^EFG_raw[t] * c^GDP[t] * k^GDP_bindG - c^EFG_GDP[t] * k^GDP_dissG tl0(42): dc^EFG_raw[t]/dt = c^EFG_GDP[t] * k^GDP_dissG - c^EFG_raw[t] * c^GDP[t] * k^GDP_bindG + c^EFG[t] * k^GTP_offG - c^EFG_raw[t] * c^GTP[t] * k^GTP_onG tl0(43): dc^EFG[t]/dt = c^EFG_raw[t] * c^GTP[t] * k^GTP_onG - c^EFG[t] * k^GTP_offG + \sum \limits_{i=1}^n c^tl_el2_EFG[i][t] * k^EFG_off - \sum \limits_{i=1}^n c^tl_el2[i][t] * c^EFG[t] * k^EFG_on

Termination.


tl0(44): dc^tl_term0[t]/dt = c^tl_el3[-1][t] * k^tRNA_diss + c^tl_term1[t] * k^RF1_off - c^tl_term0[t] * c^RF1[t] * k^RF1_on tl0(45): dc^tl_term1[t]/dt = c^tl_term0[t] * c^RF1[t] * k^RF1_on - c^tl_term1[t] * k^RF1_off + c^tl_term2[t] * k^RF3_off - c^tl_term1[t] * c^RF3[t] * k^RF3_on tl0(46): dc^tl_term2[t]/dt = c^tl_term1[t] * c^RF3[t] * k^RF3_on - c^tl_term2[t] * k^RF3_off - c^tl_term2[t] * k^protein_diss tl0(47): dc^tl_protrel[t]/dt = c^tl_term2[t] * k^protein_diss - c^tl_protrel[t] * k^30s_diss tl0(48): dc^RF1[t]/dt = c^tl_term1[t] * k^RF1_off - c^tl_term0[t] * c^RF1[t] * k^RF1_on + c^tl_protrel * k^30s_diss tl0(49): dc^RF3[t]/dt = c^tl_term2[t] * k^RF3_off - c^tl_term1[t] * c^RF3[t] * k^RF3_on + c^tl_protrel * k^30s_diss tl0(50): dc^protein[t]/dt = c^tl_term2[t] * k^protein_diss

General.


tl0(51): dc^TLATP[t]/dt = - c^AaxtRNASyn_Aaa[t] * c^ATP[t] * k^AaxtRNASyn_cat1 tl0(52): dc^TLGTP[t]/dt = c^IF2[t] * c^GDP[t] * k^IF2_deg - c^IF2_GDP[t] * c^GTP[t] * k^IF2_reg + c^EFTu_GTPEFTS[t] * k^GTP_offTu - c^EFTu_EFTS[t] * c^GTP[t] * k^GTP_onTu + c^EFG[t] * k^GTP_offG - c^EFG_raw[t] * c^GTP[t] * k^GTP_onG tl0(53): dc^TLGDP[t]/dt = c^IF2_GDP[t] * c^GTP[t] * k^IF2_reg - c^IF2[t] * c^GDP[t] * k^IF2_deg + c^EFTu_GDPEFTS[t] * k^GDP_dissTu - c^EFTu_EFTS[t] * c^GDP[t] * k^GDP_bindTu + c^EFG_GDP[t] * k^GDP_dissG - c^EFG_raw[t] * c^GDP[t] * k^GDP_bindG tl0(54): dc^Aax[t]/dt = c^AaxtRNASyn_Aax[t] * k^Aax_off - c^AaxtRNASyn[t] * c^Aax[t] * k^Aax_on tl0(55): dc^tRNA[t]/dt = c^AaxtRNASyn_AaatRNA[t] * k^tRNA_off - c^AaxtRNASyn_Aaa[t] * c^tRNA[t] * k^tRNA_on + \sum \limits_{i=1}^n c^tl_el3[i][t] * k^tRNA_dissel tl0(56): dc^TLmRNA[t]/dt = c^tl_ini0[t] * k^mRNA_off - c^30sIF3IF1[t] * c^mRNA[t] * k^mRNA_on + c^tl_protrel * k^30s_diss tl0(57): dc^50s[t]/dt = c^70sIF3[t] * k^50s_diss - c^30s[t] * c^50s[t] * k^50s_bind + c^tl_inion50s[t] * k^50s_off - c^tl_inidissIF3[t] * c^50s[t] * k^50s_on + c^tl_protrel * k^30s_diss tl0(58): dc^30s[t]/dt = c^30sIF3[t] * k^IF3_off - c^30s[t] * c^IF3[t] * k^IF3_on - c^30s[t] * c^50s[t] * k^50s_bind + c^tl_protrel * k^30s_diss

3. Diffusion<\h2>

3.1 Introduction and Motivation<\h3> In the final step of cellfree expression proteins produced are diffunding inside the microfluidic camber. We observed at the ideal case:
On the PDMS slide circle spots of bound DNA produce proteins with steadily decreasing production rate. The product is distributed homogeneously on the spot and starts diffunding freely in the cellfree mix without considering any interactions. The coated iRIf glass is an ideal sink; any proteins reaching the 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 achiev this we constructed the following system.

3.2 Model System<\h3> Effective modelling depends on a 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)

The physical process is described by the diffusion 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 and sinks inside the system. 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 Assumptions<\h3> 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: ...