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Mass action equations

This guide provides a comprehensive overview of simulating enzyme kinetics using ordinary differential equations (ODEs) in MATLAB. It includes an enzyme-substrate interaction model, defined by a series of mass action equations, and demonstrates the implementation of the model using MATLAB code. Additionally, the guide covers techniques like parameter scanning, bifurcation, and sensitivity analysis to examine how variations in rate constants affect system behavior. This tool is crucial for understanding enzyme dynamics and validating models against experimental data.

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Mass action equations

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  1. Mass action equations • E + S -(k1) E|S • E|S -(k2) E+S • E|S -(k3) E+P OR • E + S =(k1/k2)= E|S • E|S -(k3) E+P

  2. Enzyme in matlab • Matlab code and simulation/output commands % simulate a simple ode model tspan=[0 4]; [plott ploty]=ode15s(@enzyme,tspan,[1;1;0;0]); plot(plott,ploty)

  3. Enzyme simulation here… function dydt = enzyme(t,y) %----------------------------------------------------------------------- % Specify the ODE %----------------------------------------------------------------------- % Get rate constants from outside this function global xk1 xki xk2 xk1=3; xki=4; xk2=4; % Work in a more comprehensible set of variables E = y(1); S = y(2); ES = y(3); P = y(4); % Dynamic equations dydt(1) = -xk1*E*S + xki*ES + xk2*ES ; % dE/dt dydt(2) = -xk1*E*S + xki*ES ; % dS/dt dydt(3) = xk1*E*S - xki*ES - xk2*ES ; % dES/dt dydt(4) = xk2*ES ; % dP/dt dydt = dydt'; %MATLAB v5 wants a column vector

  4. Model analysis techniques • Parameter scanning • Multiple simulations at a range of values for one parameter • Bifurcation analysis • For example, identifying the point where oscillations start • Sensitivity analysis • How do the concentrations depend on one specific rate?

  5. Sensitivity analysis • Sensitivity analysis investigates the changes in the system outputs or behavior with respect to the parameter variations. It is a general technique for establishing the contribution of individual parameter values to the overall performance of a complex system. • Sensitivity analysis is an important tool in the studies of the dependence of a system on external parameters, and sensitivity considerations often play an important role in the design of control systems. • Parameter sensitivity analysis can also be utilised to validate a model’s response and iteratively, to design experiments that support the estimation of parameters Slide from Xuan Liu

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