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Solving the kinetic model and fit algorithm
This protocol is extracted from research article:
Allosteric pathway selection in templated assembly
Sci Adv, Oct 11, 2019; DOI: 10.1126/sciadv.aaw3353

We solved the kinetic model numerically using the Runge-Kutta method, with step size dt = 0.01 τB for the initial 10 τB of the simulation and dt = 0.25 τB for the remainder. Parameters for the kinetic model were found by minimizing the sum of squared residuals (R) between the kinetic diagrams and the kinetic model for each binned data pointR=i=1n[([F]ifF,i)2+([FA]ifFA,i)2+([D]ifD,i)2+([DA]ifDA,i)2](11)where n = 2000 equals the number of logarithmic spaced binned data points, [FA]i is the number density of assemblers in the FA state at data point i, and fFA, i is the concentration as obtained from the kinetic model at this data point. R was minimized using a simulated annealing fit algorithm with an exponential multiplicative cooling schedule as described by Kirkpatrick and coworkers (39). For every run of the algorithm, we performed 500 cooling steps, starting from an initial effective temperature T0 = 10, with a cooling factor of α = 0.97. At every cooling step, the seven fit parameters (kFFA, kFAF, kFD, kDF, kDDA, kDAD, and kFADA) were changed randomly using the NumPy random number generator. As the simulated annealing algorithm proceeds to lower temperatures, the tolerance for unfavorable steps decreases, lowering the acceptance rate. To ensure that sufficient meaningful parameter adjustments were made, we continuously changed the parameter adjustment step size to keep the acceptance rate at 0.5. Our use of seven fit parameters resulted in a complex goodness-of-fit landscape with many possible local minima. To find the global minimum in this landscape, we repeated the simulated annealing algorithm 100 times starting from randomized initial parameter values. Average parameter values and standard deviations were computed for the best 10 fits obtained in this way. Corresponding residuals are presented in section S5.

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