Systems biology toolbox

Numerical simulations: Time-dependent changes in state occupancies of the model in Figure 1b were evaluated by numerical integration of the resulting system of differential equations using the Systems Biology Toolbox [8 (link)] and MATLAB 2015a (MathWorks, Natick, MA, USA).
Computer assisted algebra: To evaluate the expressions in Figure 2c, we used the isAlways function contained in the symbolic toolbox of Matlab and the TrueQ function in combination with the Refine function in Mathematica.
Nonlinear constrained multivariate optimization algorithm: To minimize or maximize algebraic equations, we employed the fmincon- solver contained in Matlab. We used linear and non-linear constraints to restrict the rate constants of the model in Figure 2a to realistic values. We constrained the rates constants as follows: (i) the association rate constants for Na⁺, substrate and releaser were allowed to adopt values between 10³ M⁻¹∙s⁻¹ and 10⁸ M⁻¹∙s⁻¹ (i.e., diffusion limit); (ii) the corresponding dissociation rate constants were allowed to adopt values between 0.1 s⁻¹ and 10⁵ s⁻¹; (iii) we also constrained the affinities of Na⁺, substrate and releaser: 100 µM < [Na⁺] < 100 mM; 10 nM < [S] < 10 mM; 10 nM < [R] < 10 mM; and (iv) the rate constants for the conformational transitions were constrained to values between 0.1 s⁻¹ and 10⁵ s⁻¹.

The recorded currents were emulated with a previously published kinetic model of the transport cycle of DAT (2 (link)). This model was extended to account for binding of Zn²⁺ to DAT. The time-dependent changes in state occupancies were evaluated by numerical integration of the resulting system of differential equations using Systems Biology Toolbox (16 (link)) and Matlab 2012a (Mathworks).
The voltage-dependence of individual rates were modeled according to Laeuger (17 ) assuming a symmetric barrier as k_ij = k0_ijexp(−zQ_ijFV/2RT), with F = 96485 Cmol⁻¹, r = 8.314 JK⁻¹ mol⁻¹ and V the membrane voltage in volts and T = 293 K. The extra- and intracellular ion concentrations were set to the values used in the experiments. Substrate uptake was modeled as (TiClS × kSioff-TiCl × Si × kSion + TiClSZn × kSioff-TiClZn × Si × kSion) × NC/NA. Where NC is the number of transporters and NA is the Avogadro constant. Currents were simulated assuming a transporter density of 25 × 10⁶/cell.