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Revision as of 01:42, 28 September 2013

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Introduction

Switch modeling

• produce both states of switch

• use model to optimize switch

Mass action modeling

Definition

• differential equations

• each equation is a sum of rates

• each rate is proportional to each of its reactants

Benefits

• can minimize assumptions

• already used in chemical rate equations

• standardizes equations:

    	• x is the species vector

• a is an amount of rates changing x

    	• b is a rate constant

    	• c is a number of different species involved

    	• d is a species involved

    	• e is a number of each species involved	

    	• f is a number of species d involved

    	• g is a highest species involved

    	• l is a rate order

    	• m is the highest number of rates involved

    	• n is the number of species

Analytical modeling

• benefits

   • given parameters, it precisely predicts behavior

   • accurately predicts the data needed to determine parameters (unlike numerical analysis)

   • allows avoidance of numerical errors like rounding

   • naturally standardized

ERSESCO Algorithm

The ERSESCO algorithm the Michigan team developed provides a clearly defined method of using symbolic analysis on systems. The algorithm has seven major steps:

1. Equation

Formulate the system into a system of differential equations.

2. Reduction

Apply constraints, boundary conditions, and conservation laws to reduce the system of equations to fewer variables.

3. Solution

Solve the reduced system of equations, if possible.

4. Equilibration

Find the steady state of the system by setting differential equations to zero and solving for critical points.

5. Stabilization

Evaluate the Jacobian matrix of the system of equations at each critical point and find the corresponding eigenvalues. Eigenvalues with a negative real part are stable critical points.

6. Calibration

7. Optimization

SimBiology

• is a MatLab package provided to iGEM teams

• provides a numerical approximation

• easily creates from a diagram

Model of Recombinase Expression

The Recombinase Expression Model describes the synthesis and degradation of any protein produced at a relatively constant rate. Here, the model is used in particular to describe the synthesis and degradation reaction rates ksyn and kdeg of recombinases in the cell. The model also predicts the recombinase concentration.

Model of the Switch

The Switch Model describes how the switch equilibrates when the concentrations of the recombinases FimE and Hbif approach constant values. This model assumes that each recombinase catalyzes a one way reaction. This model predicts the degree of cooperativity n and m in the binding reaction of the recombinase to DNA for FimE and Hbif, respectively. Furthermore, this model predicts the forward catalysis rate constants, kF and KH, for FimE and Hbif.

Model of Inducible Hbif

The Inducible Hbif Model describes how the switch flips when acted upon by the Lux/HSL-controlled expression of Hbif. In this model, [HSL] represents the concentration of the species HSL:pLux:LuxR, the complex that promotes the expression of Hbif. In this model [HSL] is a function of pLux and LuxR expression levels in the cell, the amount of HSL added, and the fractional occupancies at the complexation equilibrium. It can be assumed that this complexation reaction happens instantaneously with respect to the slow, rate-limiting translation step of Hbif. This model predicts the rate k1 of Lux-induced expression of Hbif, the degradation rate kdeg of Hbif, the forward and reverse rates k2 and k-2 of Hbif catalysis on the switch, and the equilibrium constant K2 for the switch when acted upon by Hbif.

Expression Model

Equation:

Reduction: None

Solution:

Equilibration:

Stabilization:

Calibration:

Optimization:

SimBiology:

Expression Model Derivation

Switch Model

Equation:

Reduction:

Solution:

Equilibration:

Stabilization:

Calibration:

Optimization:

SimBiology:

Switch Model Derivation

Inducible Hbif Model

Equation:

Reduction:

Solution:

Equilibration:

Stabilization:

Calibration:

Optimization of Hbif:

Optimization of ON:

SimBiology:

Inducible Hbif Model Derivation

Future Directions

Data!