Figure 1. Transition rates across a... | Rockefeller University Press

Figure 1.

$Figure 1. Transition rates across a one-dimensional barrier. (A) Piecewise-harmonic bistable PMF landscape for V = 0 (Gibbs energy G(q), black curve) and V = −100 mV (W(q) = G(q) − qV, blue curve). Forward and backward rate constants are designated α and β. State coordinates (qX, ΦX); X = {R, b, P} are marked by the vertices of color-matched “discrete landscapes” (dashed lines). (B) The transmission coefficient κ as a function of friction R derived from the G landscape in A. GH, GH solution of the GLE with memory friction R(t) = Roexp(−νt)[cos(ωt) + (ν/ω)sin(ωt)], where ω/ωb = ν/ωb = 0.1 (Grote and Hynes, 1980); Kr, Kramers’ intermediate-to-large friction equation; LF, LF approximation in the Smoluchowski regimen; MM, Mel’nikov and Meshkov exact solution of Langevin equation encompassing small friction (Mel’nikov and Meshkov, 1986). The GH curve was obtained numerically by finding the positive root λ of ϑ−1=s2+γ^s−ωb2 consistent with γ^(s)≡R^(s)/L = (Ro/L)(s + 2ν)/[(s + ν)2 + ω2] and plotting κ = λ/ωb against the zero-frequency friction R^(0) = 2Roν/(ν2 + ω2). The value of the TST rate constant is αTST = 71 ms−1 for L = 10−15 mV · ms2/eo (= 9.7 kDÅ2/eo2).$

Transition rates across a one-dimensional barrier. (A) Piecewise-harmonic bistable PMF landscape for V = 0 (Gibbs energy G(q), black curve) and V = −100 mV (W(q) = G(q) − qV, blue curve). Forward and backward rate constants are designated α and β. State coordinates (q_X, Φ_X); X = {R, b, P} are marked by the vertices of color-matched “discrete landscapes” (dashed lines). (B) The transmission coefficient κ as a function of friction R derived from the G landscape in A. GH, GH solution of the GLE with memory friction R(t) = R_oexp(−νt)[cos(ωt) + (ν/ω)sin(ωt)], where ω/ω_b = ν/ω_b = 0.1 (Grote and Hynes, 1980); Kr, Kramers’ intermediate-to-large friction equation; LF, LF approximation in the Smoluchowski regimen; MM, Mel’nikov and Meshkov exact solution of Langevin equation encompassing small friction (Mel’nikov and Meshkov, 1986). The GH curve was obtained numerically by finding the positive root λ of $ϑ^{- 1} = s^{2} + \hat{γ} s - ω_{b}^{2}$ consistent with $\hat{γ} (s) \equiv \hat{R} (s) / L$ = (R_o/L)(s + 2ν)/[(s + ν)² + ω²] and plotting κ = λ/ω_b against the zero-frequency friction $\hat{R} (0)$ = 2R_oν/(ν² + ω²). The value of the TST rate constant is α_TST = 71 ms⁻¹ for L = 10⁻¹⁵ mV · ms²/e_o (= 9.7 kDÅ²/e_o²).

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