Given a fixed number of rate categories k, the parameters of the discrete substitution model is a vector of rates r = (r1, r2, …, r k ) and a vector of rate-class assignments for each lineage c = (c1, c2, …, c2 n − 2) where c i ∈ 1 … k.
Furthermore, we define an educt stoichiometry matrix Se, which only contains the negative values from S. The reaction rate laws v can be written as v x, k) = diag(k) c(x), where k := (k1,..., kr) represents a vector of elementary rate constants and c j (x ) : = ∏ i = 1 m x i − s i j e, j = 1,..., r contains the products of substrate concentrations according to the law of mass action[ 36].
The right-hand side of Eq. (1) can be expressed as the stoichiometric matrix S ∈ ℝ n × q times a vector of reaction rates r = r x, p, u, t); r ∈ ℝ q ẋ = S r (x, p, u, t ).
Traditional search methods generate a full vector of rate parameters, simulate the model with this full set of parameters and then accept, reject or adjust the parameters based on how well the simulation agrees with experimental measurements.
where M [xi] is a vector of age-specific mortality rates with elements m' x for x < = x i and m x for x > =x i.
For each neuron, we first collected a vector of trial-averaged firing rates over time for the forced left condition.
In addition, x1j is a vector of variables affecting the hazard rate of returning to accommodated work (j =1), non-accommodated work (j = 2) and to a new employer (j = 3), and β1j; j = 1, 2, 3 is a corresponding row vector of regression coefficients.
The ordinary differential equations (ODE) model of a mass-action biochemical mechanism (1) can be written in vector form as (4) u ˙ (t ) = Sw (u ), where u(t)= u1 t),…, u n (t)) T is the concentration vector of the chemical species of (1), S ji = β ji − α ji are the entries of the stoichiometric matrix S and w u)= w1 u),…, w m (u)) T is the vector of rate functions (3).
where j is the Jacobian matrix that consists of the first derivatives ofthe network errors, e is a vector of network errors, μ isthe learning rate, and I is the identity matrix.
Here c ext is a vector of concentrations of extracellular metabolites, K ext a matrix of stoichiometric coefficients, D is the dilution rate, u ext is a vector of volumetric feeding rates, and r ext is the kinetic rate vector.
v 13C is a vector of intracellular fluxes obtained from C-based MFA, and v fmt is a vector of the cell growth rate and glucose uptake rate from the continuous cell culture.
