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Team:ZJU-China/Modeling/Biobrick
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| + | <p>Secondly,in the anaerobic condition, that is, when the expression of promoter vgb could be | ||
| + | neglected, we set<strong>α</strong>to be zero, and set<strong>β</strong>to be bigger than the dissociation constant in (13). | ||
| + | Then we have</p> | ||
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Revision as of 12:32, 3 October 2011
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Modeling|biobrick |
This model is used for simulating the behavior of three genetic circuits we designed
Overview
Modeling
Results
Introduction
abstract:
We will model our gene regulatory networks using Michaelis-Menten enzymatic kinetics,together with the usual rules of reaction kinetics. The resulting models, when spatial
effects are neglected, are given in terms of ordinary differential equations describing the
rate of change of the concentrations of gene products and proteins. A key component of all
these models is the Hill function, used to describe the transcription phase. The presence of
this highly nonlinear function, whilst accurately modeling the network, inevitably leads to
restrictions on the analytical tools available to understand and predict the
dynamics.
Basic concepts and assumptions
The ODE formalism models the concentrations of RNAs, proteins, and other molecules by
time-dependent variables with values contained in the set of nonnegative real numbers.
Regulatory interactions take the form of functional and differential relations between the
concentration variables. For a typical transcription-translation process, the ODEs modeling
approach associates two ODEs with any given gene i; one modeling the rate of change of the
concentration of the transcribed mRNA r_i, and the other describing the rate of change of
the concentration of its corresponding translated protein p_i. Thus for our network with 3
genes we have:
Where (1) describes transcription, (2) describes translation, and i = 1,…,N. The functions R_i(p_j)
describe the dependence of mRNA concentration on protein concentration p_j (If protein p_j has
no effect on mRNA r_i, then correspond function is set to zero.) The functional F(·) in (1) is
defined in terms of sums and products of functions R_i. Function P_i in (2) describes the
translation of the mRNA r_i into a protein p_i. Parameters γ_i, δ_i (i = 1,…,N), represent the
degradation parameters of the mRNAs and proteins produced by gene i. As is common, we shall
assume that the degradation of proteins or mRNAs is not regulated, namely that it does not
depend on the concentrations of other molecules in the cell. Function R_i is assumed to be in the
form of Hill function as usual (since our cases are all inhibitors, we shall denote the Hill function
h-(p,K,n)), and the function P_i is taken to be a linear term proportional to the concentration of
mRNA r_i.
where K_i is the microscopic dissociation constant, and n_i is Hill coefficient, describing cooperativity.
Equations&Parameters
Based on the above, our model could be founded with following ODEs:
Where
Parameters
Variables are defined in following table.
We should note that α and β are two functions of the oxygen concentration, which are determined by the properties of corresponding promoter. For a certain depth of the biofilm, the concentration of oxygen is a constant in our model, so are and . Therefore we could solve these equations at different oxygen concentration and combine all the results to show how this system work.
Because of the nonlinearity of the Hill functions, the solutions of a system of ordinary differential equations of a network of many genes cannot generally be determined by analytical means.
Equilibrium analysis
Notice that we care more about the final state of the bacteria in different depth of the
biofilm(thus in different oxygen condition), we assume that the system has reached a steady
state. We could calculus this steady state by setting all derivatives with respect to time to zero.
That is, let
With (4)-(10), we have:
Clearly, the behavior of this system is determined by the two functions, thus determined by the behavior of promoter vgb & promoter fdhF. On the other hand, the solutions of equations The two functions αand β are hard to determine precisely, but their behavior could be illustrated in follow graph qualitatively.
First case, When the expression of promoter vgb and fdhF could be neglected, that is, we could
setαandβto be both zero. The ODEs system become
The results are shown as follow.
Secondly,in the anaerobic condition, that is, when the expression of promoter vgb could be neglected, we setαto be zero, and setβto be bigger than the dissociation constant in (13). Then we have
