IGEM:Cambridge/2008/Notebook/Modelling
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  +  = Modelling ReactionDiffusion Systems =  
  +  [[Image:TuringPattern1.png  Pulsating spots  thumb  right ]]  
  +  
  +  Mathematically, reactiondiffusion systems are coupled nonlinear differential equations that can be solved numerically. Our first step will be to implement/model a simple twocomponent system originally proposed by Turing. After validating our numerical method and computating our first patterns (img to the right), we will be thinking about a more realistic system describing the behaviour of the activator and inhibitor system that we intend to engineer with B.subtilis. This will include an analysis of enzyme kinetics and we hope to deduce the parameter ranges, in which our B.subtilis construct will be able to form patterns, thus feeding back into our design decisions (promoter strength, rbs choice etc.).  
+  We will also take an investigative approach and ask whether Turinglike patterns can originate from a system that is simpler than the one envisioned, consisting of fewer components, e.g. depending on a single signalling molecule only.  
  +  == Introduction ==  
  +  
  +  
  +  
+  How does pattern formation occur? Turing considered diffusion to be the crucial component. He showed that pattern formation occurs of the system has a stable steady state in the absence of diffusion, but allows unstable states to develop when the diffusion term is added. This results in divergence (or, in a biological context, gene expression/cell differentiation) and subsequent pattern formation. In the following sections, we shall deduce conditions for pattern formation for this simple system.  
+  
+  We will work with a twocomponent system and we nondimensionalise the variables. The resulting general form is:  
+  
+  :<math>\frac{\partial A}{\partial t} = \gamma f(A,B) + \nabla^2 A</math>  
+  
+  :<math>\frac{\partial B}{\partial t} = \gamma g(A,B) + d \nabla^2 B</math>  
+  
+  where <math>\gamma</math> (determining the scale) is a constant and d stands for the diffusion ratio. We note that nondimensionalisation has the added advantage that we can now map a specific pattern onto a wide range of biological parameters, the easiest example being that two different pairs of diffusion rates will result in the same pattern if the respective diffusion ratios remain unchanged.  
+  
+  == Turing System ==  
+  (also known as Schnakenberg reaction)  
+  
+  === Case: Without diffusion, need stable steady state ===  
+  
+  [[Image:Turing_moving.png  Moving flame fronts  thumb  right]]  
+  
+  Consider the following system without diffusion terms:  
+  
+  :<math>\frac{\partial A}{\partial t} = \gamma (\alpha  A + A^2B) = \gamma f(A,B)</math>  
+  :<math>\frac{\partial B}{\partial t} = \gamma (\beta  A^2B) = \gamma g(A,B)</math>  
+  
+  We determine the steady state solution and add a small pertubation <math>\tilde{A}, \tilde{B} </math> to linearise the system about the steady state in order to determine its stability. In Matrix notation, the linearised system can be written as:  
+  
+  :<math>  
+  \begin{pmatrix}  
+  \tilde{A_t} \\  
+  \tilde{B_t} \\  
+  \end{pmatrix}  
+  =  
+  \gamma  
+  \begin{pmatrix}  
+  f_A & f_B \\  
+  g_A & g_B \\  
+  \end{pmatrix}  
+  .  
+  \begin{pmatrix}  
+  \tilde{A} \\  
+  \tilde{B} \\  
+  \end{pmatrix}  
+  </math>  
+  
+  where the Jacobian is evaluated at the steady states of A and B. This is a set of coupled firstorder ODEs and solutions are proportional to <math>exp(\lambda t)</math>. In order to have a stable steady state, we need the real part of <math>\lambda</math> to be negative.  
+  
+  This requires <math>f_A + g_B < 0</math> (need ve trace) and <math>f_A g_Bf_B g_A > 0</math> (need +ve determinant), which we note as our first two conditions for pattern formation.  
+  
+  === Case: With diffusion, need unstable steady states ===  
+  Now we add diffusion to our system and require the resulting steady state to be unstable. After linearisation, the equation can be expressed as follows:  
+  
+  :<math>  
+  \begin{align}  
+  \begin{pmatrix}  
+  \tilde{A_t} \\  
+  \tilde{B_t} \\  
+  \end{pmatrix}  
+  &=  
+  \gamma  
+  \begin{pmatrix}  
+  f_A & f_B \\  
+  g_A & g_B \\  
+  \end{pmatrix}  
+  .  
+  \begin{pmatrix}  
+  \tilde{A} \\  
+  \tilde{B} \\  
+  \end{pmatrix}  
+  
+  +  
+  
+  \begin{pmatrix}  
+  1 & 0 \\  
+  0 & d \\  
+  \end{pmatrix}  
+  .  
+  \nabla^2  
+  \begin{pmatrix}  
+  \tilde{A} \\  
+  \tilde{B} \\  
+  \end{pmatrix}  
+  \\  
+  & \equiv \gamma \mathbf{M}.\mathbf{w} + \mathbf{D}.\nabla^2\mathbf{w}  
+  \end{align}  
+  </math>  
+  
+  In the following steps, we consider solutions of this particular form <math>\mathbf{w}(\mathbf{x},t) = \sum_{k} c_{k} e^{\lambda t} \mathbf{v}_k(\mathbf{x}) </math>  
+  
+  with <math> \mathbf{v}_k </math> satisfying <math> \nabla^2 \mathbf{v}_k + k^2\mathbf{v}_k = 0</math>.  
+  
+  
+  After substituting into the linearised system, we can see that for each index <b>k</b>, we require:  
+  
+  :<math>det(\lambda \mathbf{I}  \gamma \mathbf{M} + k^2\mathbf{D}.\mathbf{v}_k) = 0</math>.  
+  
+  
+  We now need to determine the conditions for unstable steady states, i.e. conditions on the parameters for which Re(<math>\lambda</math>)>0 hold. The calculations will be standard and similar to the previous case: We need the trace and the determinant of the matrix <math>\gamma \mathbf{M}k^2\mathbf{D.v}_k</math> to infer stability properties. Thus, we arrive at a set of inequalities that we can simplify to give us the following conditions:  
+  
+  <!  
+  :<math>  
+  \begin{align}  
+  dk^4\gamma k^2(df_u+g_v)+\gamma^2(f_A g_Bg_A f_B) &< 0 \\  
+  \gamma (f_A + g_B)  k^2(1+d) &> 0 \\  
+  f_A+g_B &< 0 \\  
+  f_A g_B  f_A g_B &> 0\\  
+  \end{align}  
+  </math>  
+  
+  with the last two lines being required by the previous case without diffusion. After substituting f and g (i.e. their respective derivatives evaluated at the steady state) into these inequalities we will arrive at a further set of inequalities involving the parameters <math>\alpha, \beta, d</math> only:  
+  >  
+  :<math>  
+  \begin{align}  
+  0 < \beta  \alpha &< (\alpha+\beta)^3 \\  
+  d(\beta\alpha) &> (\alpha+\beta)^3 \\  
+  (d(\beta\alpha)(\alpha+\beta)^3)^2 &> 4d(\alpha+\beta)^4\\  
+  \end{align}  
+  </math>  
+  
+  
+  These are a set of conditions that our parameters must satisfy in order for this particular system to develop patterns.  
+  
+  Given a (twocomponent) system, we can now infer which parameters to choose in order for the system to develop patterns. This saves us a lot of time as reaction diffusion systems are extremely sensitive to parameter changes and simply trying out sets of parameters will usually result in no patterning at all.  
+  
+  =Bacillus and its signalling systems=  
+  
+  We will first look at the specific components of our construct and models describing the behaviour of the agr and lux quorumsensing systems will be introduced. We will then model our envisioned activatorinhibitor system and deduce under which conditions pattern formation will occur.  
+  <!  
+  As a (temporary) simplifying assumption, we will use the following parameter values: 10, 1 and 0.1 for high, medium and low rates with units sec<sup>1</sup>. Variables in brackets denote concentrations of the corresponding enzyme/enzyme complex/chemical player. We match those values (ironically) by considering biological factors, e.g. one can infer that phosphorylation rates (give it a 10) are much higher than constitutive expression rates (give it a 1 or even 0.1 for low levels of constitutive expression) etc.  
+  
+  Once the models are in place, we shall work with the actual biological parameters, given they can be found in literature or inferred from wetwork.  
+  
+  >  
+  ==Peptidesignalling system==  
+  
+  === Receiver ===  
+  
+  We have created a simple model describing the agrreceiver device and it exhibits a twostate behaviour, akin to a switch. This is typical for quorumsensing systems: once concentrations of AIP (our signalling molecule) pass a certain threshold value, the agrreceiver will jump from its previously "off"state (negligible levels of PoPS from the P2 promoter) into an "on"state (significant levels of PoPS). In nature, once the bacterial density (and thus AIP concentration) surpasses a critical value, the bacteria will e.g. sporulate, become virulent or fluoresce  behaviour that is biologically expensive (however, critical) and only advantageous at high cell densities.  
+  
+  The equations of this model read as follows:  
+  
+  :<math>  
+  \begin{align}  
+  \frac{\partial [A]}{\partial t} &= p_{2} \frac{[A^{+}]^n}{1+[A^{+}]^n} + \beta  p_{+} [CP] [A] + p_{} [A^{+}]  \delta [A] \\  
+  \frac{\partial [A^{+}]}{\partial t} &= p_{+} [CP] [A]  p_{} [A^{+}]  \delta [A^{+}] \\  
+  \frac{\partial [C]}{\partial t} &= p_{2} \frac{[A^{+}]^n}{1+[A^{+}]^n} + \beta  c_{+} [C] [P] + c_{} [CP] [P]  \delta [C] \\  
+  \frac{\partial [P]}{\partial t} &=  c_{+} [C] [P] + c_{} [CP]  \delta [P] \\  
+  \frac{\partial [CP]}{\partial t} &= c_{+} [C] [P]  c_{} [CP]  \delta [CP] \\  
+  \end{align}  
+  </math>  
+  
+  A rigorous justification (i.e. all the steps leading to this set of equations) should probably be on here as well.  
+  
+  === Sender ===  
+  Interesting note: There is experimental evidence that AIP production rates are not depended on the basal expression level of agrB and agrD.  
+  
+  ==AHLsignalling system==  
+  
+  According to [http://openwetware.org/wiki/IGEM:Cambridge/2008/Turing_Pattern_Formation/Experiments#Testing_degradation_of_AHL_by_bacillus_aiiA our experiments], B.subtilis does not degrade AHL.  
+  
+  Model behaviour, look at Bangalore 2007 & Canton et al. (2008), assay luxsystem in B.subtilis.  
+  
+  ==A Repressible promoter==  
+  
+  ==Activatorinhibitor system==  
+  Local activation, lateral inhibition.  
+  
+  =Beyond Turing=  
+  
+  Pattern formation with a single signalling molecule only?  
+  
+  
+  }  
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Modelling ReactionDiffusion SystemsMathematically, reactiondiffusion systems are coupled nonlinear differential equations that can be solved numerically. Our first step will be to implement/model a simple twocomponent system originally proposed by Turing. After validating our numerical method and computating our first patterns (img to the right), we will be thinking about a more realistic system describing the behaviour of the activator and inhibitor system that we intend to engineer with B.subtilis. This will include an analysis of enzyme kinetics and we hope to deduce the parameter ranges, in which our B.subtilis construct will be able to form patterns, thus feeding back into our design decisions (promoter strength, rbs choice etc.). We will also take an investigative approach and ask whether Turinglike patterns can originate from a system that is simpler than the one envisioned, consisting of fewer components, e.g. depending on a single signalling molecule only. IntroductionHow does pattern formation occur? Turing considered diffusion to be the crucial component. He showed that pattern formation occurs of the system has a stable steady state in the absence of diffusion, but allows unstable states to develop when the diffusion term is added. This results in divergence (or, in a biological context, gene expression/cell differentiation) and subsequent pattern formation. In the following sections, we shall deduce conditions for pattern formation for this simple system. We will work with a twocomponent system and we nondimensionalise the variables. The resulting general form is: where γ (determining the scale) is a constant and d stands for the diffusion ratio. We note that nondimensionalisation has the added advantage that we can now map a specific pattern onto a wide range of biological parameters, the easiest example being that two different pairs of diffusion rates will result in the same pattern if the respective diffusion ratios remain unchanged. Turing System(also known as Schnakenberg reaction) Case: Without diffusion, need stable steady stateConsider the following system without diffusion terms: We determine the steady state solution and add a small pertubation to linearise the system about the steady state in order to determine its stability. In Matrix notation, the linearised system can be written as: where the Jacobian is evaluated at the steady states of A and B. This is a set of coupled firstorder ODEs and solutions are proportional to exp(λt). In order to have a stable steady state, we need the real part of λ to be negative. This requires f_{A} + g_{B} < 0 (need ve trace) and f_{A}g_{B} − f_{B}g_{A} > 0 (need +ve determinant), which we note as our first two conditions for pattern formation. Case: With diffusion, need unstable steady statesNow we add diffusion to our system and require the resulting steady state to be unstable. After linearisation, the equation can be expressed as follows: In the following steps, we consider solutions of this particular form with satisfying .
Given a (twocomponent) system, we can now infer which parameters to choose in order for the system to develop patterns. This saves us a lot of time as reaction diffusion systems are extremely sensitive to parameter changes and simply trying out sets of parameters will usually result in no patterning at all. Bacillus and its signalling systemsWe will first look at the specific components of our construct and models describing the behaviour of the agr and lux quorumsensing systems will be introduced. We will then model our envisioned activatorinhibitor system and deduce under which conditions pattern formation will occur. Peptidesignalling systemReceiverWe have created a simple model describing the agrreceiver device and it exhibits a twostate behaviour, akin to a switch. This is typical for quorumsensing systems: once concentrations of AIP (our signalling molecule) pass a certain threshold value, the agrreceiver will jump from its previously "off"state (negligible levels of PoPS from the P2 promoter) into an "on"state (significant levels of PoPS). In nature, once the bacterial density (and thus AIP concentration) surpasses a critical value, the bacteria will e.g. sporulate, become virulent or fluoresce  behaviour that is biologically expensive (however, critical) and only advantageous at high cell densities. The equations of this model read as follows: A rigorous justification (i.e. all the steps leading to this set of equations) should probably be on here as well. SenderInteresting note: There is experimental evidence that AIP production rates are not depended on the basal expression level of agrB and agrD. AHLsignalling systemAccording to our experiments, B.subtilis does not degrade AHL. Model behaviour, look at Bangalore 2007 & Canton et al. (2008), assay luxsystem in B.subtilis. A Repressible promoterActivatorinhibitor systemLocal activation, lateral inhibition. Beyond TuringPattern formation with a single signalling molecule only?
