Estimating kinetic mechanisms with prior knowledge I: Linear parameter constraints

Abstract

New mathematical tools are needed to incorporate existing knowledge into kinetic models of ion channels and other proteins. Salari et al. describe an algebraic transformation that can enforce linearly interdependent parameters into kinetic models in order to test new hypotheses.

Estimating kinetic mechanisms with prior knowledge II: Behavioral constraints and numerical tests

“…A theoretical background was given in part one of this study ( Salari et al, 2018 ), where we briefly introduced a few prerequisite topics (kinetic mechanisms, model topology, and parameter estimation). Then, we presented a mathematical formalism and computational procedure for enforcing linear parameter constraints.…”

“…Likewise, we cannot use that formalism to solve any parameter constraint that cannot be written as a linear relationship between the transformed model parameters, as captured by the generalized linear constraint Eqs. 32 and 33 ( Salari et al, 2018 ). Instead, the solution we propose here for handling behavioral constraints and arbitrary parameter relationships is to include them into the cost function.…”

“…Another example is parameterizing an exponential factor as a product of more than one variable: where a and b could stand for the δ ij and z ij parameters in Eq. 2 ( Salari et al, 2018 ) and C is a constant equal to F /( R × T ). Likewise, this equation cannot be handled with the formalism from part one because it is nonlinear.…”

“…Thus, the above constraints could be rewritten as follows: The cost function components F C that correspond to the above equality and inequality relationships can be formulated as follows: where α is a weighting factor with the following properties: and where “constraint” refers to the left-side term of a constraint equation (Eq. 37; Salari et al, 2018 ). Thus, these cost function components are equal to zero when the underlying constraints are exactly satisfied but take a positive and quadratically increasing value when the constraint relationships are not satisfied.…”

“… and represent rate constant parameters, as defined by Eq. 1 from the companion paper ( Salari et al, 2018 ; rate constant ), a 1 is an allosteric factor, and N C is the channel count. P o and f R represent model properties, as defined in Fig.…”