Estimation of rate constants

This protocol is extracted from research article:

High-speed AFM reveals accelerated binding of agitoxin-2 to a K+ channel by induced fit

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Sci Adv**,
Jul 3, 2019;
DOI:
10.1126/sciadv.aax0495

High-speed AFM reveals accelerated binding of agitoxin-2 to a K+ channel by induced fit

Procedure

According to the event-oriented analysis, there are two conformational states in the channel: high- and low-affinity states, denoted as C_{H} and C_{L}, respectively. Considering the binding of AgTx2, four states were expected to exist: C_{H} (state 1; this numbering was used in the mathematical treatment below), C_{L} (state 2), C_{H}Tx (state 3), and C_{L}Tx (state 4) (Fig. 4A). The four-state model was also supported by the conventional dwell time analysis, in which both bound and unbound dwell time distribution could be fitted by double-exponential functions (fig. S2, A and B). (A three-state kinetic model was also tested but did not successfully reproduce the measurements: In particular, the model could not reproduce a sudden decrease of *P*_{bound} when *t*_{persist} = 0 s in Fig. 3E since the model lacks C_{L}Tx.) The rate from state *i* to *j* is denoted by *k _{ij}*. The time courses of the probabilities of states were calculated by solving the following simultaneous differential equation$$\begin{array}{c}\frac{{\mathit{dP}}_{1}}{\mathit{dt}}=-({k}_{12}+{k}_{13}[\text{AgTx}2]){P}_{1}+{k}_{21}{P}_{2}+{k}_{31}{P}_{3}\\ \frac{{\mathit{dP}}_{2}}{\mathit{dt}}={k}_{12}{P}_{1}-({k}_{21}+{k}_{24}[\text{AgTx}2]){P}_{2}+{k}_{42}{P}_{4}\\ \frac{{\mathit{dP}}_{3}}{\mathit{dt}}={k}_{13}[\text{AgTx}2]{P}_{1}-({k}_{31}+{k}_{34}){P}_{3}+{k}_{43}{P}_{4}\\ \frac{{\mathit{dP}}_{4}}{\mathit{dt}}={k}_{24}[\text{AgTx}2]{P}_{2}+{k}_{34}{P}_{3}-({k}_{42}+{k}_{43}){P}_{4}\end{array}$$where

To estimate the rate constants, four time courses of binding probability were used to fit simultaneously: *P*_{bound}(Δ*t*; *t*_{persist} = 0 s), *P*_{bound}(Δ*t*; *t*_{persist} = 1.0 s), *P*_{unbound}(Δ*t*; *t*_{persist} = 0 s), and *P*_{unbound}(Δ*t*; *t*_{persist} = 1.5 s). *P*_{bound}(Δ*t*; *t*_{persist}) did not greatly alter *t*_{persist} > 1.0 s, and *P*_{unbound}(Δ*t*; *t*_{persist}) did not greatly change *t*_{persist} > 1.5 s. *P*_{3} plus *P*_{4} [*P*_{3 + 4}(Δ*t*)] is the binding probability, which is measured by AFM.

The rates were optimized using the least-squares method, in which the following function (*E*) was minimized$$\begin{array}{cc}\hfill E=& \int d(\mathrm{\Delta}t)w(\mathrm{\Delta}t)({({P}_{\text{bound}}(\mathrm{\Delta}t;{t}_{\text{persist}}=0)-{P}_{3+4}(\mathrm{\Delta}t))}^{2}\hfill \\ & +{({P}_{\text{bound}}(\mathrm{\Delta}t;{t}_{\text{persist}}=1.0\mathrm{s})-({s}_{1}{P}_{3+4}(\mathrm{\Delta}t)+1-{s}_{1}))}^{2}\hfill \\ & +{({P}_{\text{unbound}}(\mathrm{\Delta}t;{t}_{\text{persist}}=0)-{P}_{3+4}(\mathrm{\Delta}t))}^{2}\hfill \\ & +{({P}_{\text{unbound}}(\mathrm{\Delta}t;{t}_{\text{persist}}=1.5\mathrm{s})-{s}_{2}{P}_{3+4}(\mathrm{\Delta}t))}^{2})\hfill \end{array}$$where weights *w* of 10 for Δ*t* < 0.5 s, 2 for 0.5 s ≤ Δ*t* < 5 s, and 1 otherwise were used to precisely fit the sudden changes on the short time scale. In addition, two scaling parameters (*s*_{1} = 0.9 and *s*_{2} = 0.75) were used to determine the best fit. These four curves were fitted by solving the above simultaneous differential equations with different initial conditions of the probability: *P*_{bound}(Δ*t*; *t*_{persist} = 0 s) was calculated by setting *P*_{4}(0) = 0.65 and *P*_{3}(0) = 0.35. *P*_{bound}(Δ*t*; *t*_{persist} = 1.0 s) was calculated by setting *P*_{3}(0) = 1, assuming all channels adopt a high-affinity state if a long time passes after the binding of AgTx2. Similarly, *P*_{unbound}(Δ*t*; *t*_{persist} = 1.5 s) was calculated by setting *P*_{2}(0) = 1, assuming that all channels adopt a low-affinity state if a long time passes after the dissociation of AgTx2. *P*_{unbound}(Δ*t*; *t*_{persist} = 0 s) was calculated by setting *P*_{1}(0) = 0.35 and *P*_{2}(0) = 0.65. All the rates were limited to less than 100 s^{−1} in the fitting because AFM cannot measure binding events faster than 100 s^{−1}; AFM requires at least 10 ms to scan the entire surface of the AgTx2 molecule. The resultant fitted curves, in addition to the measured ones, are shown in fig. S2 (C and D). The rates were evaluated at 20 nM AgTx2 because the largest amount of data was obtained at that concentration.

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