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A minimization principle for transition paths of maximum flux for collective variables

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Considered is the construction of transition paths of conformational changes for proteins and other macromolecules, using methods that do not require the generation of dynamics trajectories. Special attention is given to the use of a reduced set of collective variables for describing such paths. A favored way to define transition paths is to seek channels through the transition state having cross sections with a high reactive flux (density of last hitting points of reactive trajectories). Given here is a formula for reactive flux that is independent of the parameterization of “collective variable space.” This formula is needed for the principal curve of the reactive flux (as in the revised finite temperature string method) and for the maximum flux transition (MaxFlux) path. Additionally, a resistance functional is derived for narrow tubes, which when minimized yields a MaxFlux path. A strategy for minimization is outlined in the spirit of the string method. Finally, alternative approaches based on determining trajectories of high probability are considered, and it is observed that they yield paths that depend on the parameterization of collective variable space, except in the case of zero temperature, where such a path coincides with a MaxFlux path.

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Correspondence to Robert D. Skeel.

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This material is based upon work supported by Grant R01GM083605 from the National Institute of General Medical Sciences.



1.1 Area metric

A formula is derived for measuring surface area with a metric tensor \(G(\zeta )\). An infinitesimal surface \(\delta \varSigma\) enclosing a point \(\zeta _0\) can be expressed as the intersection of an infinitesimal region \(\delta \varOmega\) surrounding \(\zeta _0\) and the plane \(\hat{n}\cdot (\zeta -\zeta _0) = 0\). To determine the area of \(\delta \varSigma\), introduce variables \(\zeta ' = G^{1/2}(\zeta -\zeta _0)\), for which distances and angles are locally Euclidean. In the new coordinates, the surface becomes the intersection of some transformed region \(\delta \varOmega '\) and the plane \(\hat{n}'\cdot \zeta ' = 0\) where \(\hat{n}' = G^{-1/2}\hat{n}/|G^{-1/2}\hat{n}|\) and \(|{\delta \zeta }| = ({\delta \zeta }^{\mathsf {T}}{\delta \zeta })^{1/2}\). Its surface area is

$$\begin{aligned} \int _{\delta \varSigma '}{\mathrm {d}} S_{\zeta '}= & {} \int _{\delta \varOmega '}\delta (\hat{n}'\cdot \zeta '){\mathrm {d}}\zeta ' \\= & {} \int _{\delta \varOmega } \delta (\hat{n}'\cdot G^{1/2}(\zeta -\zeta _0))\det G^{1/2} {\mathrm {d}}\zeta \\= & {} \int _{\delta \varOmega } \delta ((\hat{n}/|G^{-1/2}\hat{n}|)\cdot (\zeta -\zeta _0))(\det G)^{1/2} {\mathrm {d}}\zeta \\= & {} \int _{\delta \varOmega } |G^{-1/2}\hat{n}|\delta (\hat{n}\cdot (\zeta -\zeta _0))(\det G)^{1/2} {\mathrm {d}}\zeta \\= & {} \int _{\delta \varSigma }|G^{-1/2}\hat{n}|(\det G)^{1/2}{\mathrm {d}} S_\zeta \end{aligned}$$

where the first and last equality follow by applying the coarea formula This result is restated in Eq. (8) using the collective variable metric.

1.2 Path of maximum reactive flux

Define \({\hat{q}}(s) = q(Z(s))\), and define \(\sigma (\zeta )\) implicitly by \(q(\zeta ) = {\hat{q}}(\sigma (\zeta ))\). The assumption Eq. (10) that isocommittors are planar implies that \(\sigma (\zeta )\) satisfies

$$\begin{aligned} \hat{n}(\sigma (\zeta ))^{\mathsf {T}}(\zeta - Z(\sigma (\zeta ))) = 0, \end{aligned}$$

so on each isocommittor \(\varPi (s)\),

$$\begin{aligned} \nabla \sigma (\zeta )|_{\sigma (\zeta ) = s} = \frac{ \hat{n}(s)}{Z_s(s)^{\mathsf {T}}\hat{n}(s) -\hat{n}_s(s)^{\mathsf {T}}(\zeta - Z(s))}. \end{aligned}$$

 It is convenient to use Eq. (12) to write Eq. (9) as

$$\begin{aligned} j(\zeta ) = d_0 C_\xi \gamma (\zeta )^{-1}c(\zeta , G(\zeta )^{-1}\nabla q(\zeta )). \end{aligned}$$

Using Eq. (29) and \(q(\zeta ) = {\hat{q}}(\sigma (\zeta ))\), Eq. (30) becomes

$$\begin{aligned} j(\zeta )|_{\sigma (\zeta ) = s} = d_0 C_\xi {\hat{q}}_s(s) \frac{\gamma (\zeta )^{-1}c(\zeta , G(\zeta )^{-1}\hat{n}(s))}{\hat{n}(s)^{\mathsf {T}}Z_s(s) -\hat{n}_s(s)^{\mathsf {T}}(\zeta - Z(s))}. \end{aligned}$$

Maximizing \(j(\zeta )\) over \(\zeta\) subject to \(\hat{n}(s)^{\mathsf {T}}(\zeta - Z(s)) = 0\) gives the condition

$$\begin{aligned} -\frac{\nabla _\zeta \gamma (Z)}{\gamma (Z)} -\frac{(\nabla _\zeta c)(Z, G(Z)^{-1}\hat{n})}{c(Z, G(Z)^{-1}\hat{n})} +\frac{\hat{n}_s}{\hat{n}^{\mathsf {T}}Z_s}~\Vert ~\hat{n} \end{aligned}$$

for the value Z(s) that maximizes \(j(\zeta )\). (See Ref. [27], Sec. 3.4.)

The assumption that the flux J(Z) is parallel to the direction \(Z_s\) of the path implies \(G(Z)^{-1}\hat{n}~\Vert ~Z_s\). Noting that \(\nabla _\omega c(\zeta ,\omega ) = G(\zeta )\omega /c(\zeta ,\omega )\), condition (31) becomes

$$\begin{aligned} -\frac{\nabla _\zeta \gamma (Z)}{\gamma (Z)} -\frac{\nabla _\zeta c(Z, Z_s)}{c(Z, Z_s)} +\frac{(\nabla _\omega c(Z, Z_s))_s}{c(Z, Z_s)}~\Vert ~\nabla _\omega c(Z, Z_s). \end{aligned}$$

(See Ref. [27], Eq. (10).) This can be written as an equation by expressing the left-hand side as an undetermined scalar times the right-hand side and then solving for the scalar after premultiplying this equation by \(Z_s^{\mathsf {T}}\). The result is Eq. (11).

1.3 Narrow tube flow rate formula

Using Eq. (6) and Eq. (18) of Ref. [18], the flow rate can be expressed as a volume integral:

$$\begin{aligned} \text{ flow } \text{ rate } =\int _\varOmega \rho _\xi (\zeta )\nabla q(\zeta )^{\mathsf {T}}D(\zeta )\nabla q(\zeta ){\mathrm {d}}\zeta = C_\xi d_0 I(q). \end{aligned}$$

Foliating \(\varOmega\) in Eq. (6) gives

$$\begin{aligned}I(q) = \int _0^1 \int _\varOmega {\mathrm {e}}^{-\beta F(\zeta )} \nabla q(\zeta )^{\mathsf {T}}G(\zeta )^{-1}\nabla q(\zeta )\delta (\sigma (\zeta ) - s) {\mathrm {d}}\zeta \,{\mathrm {d}} s \end{aligned}$$

where the decomposition \(\hat{q}(\sigma (\zeta )) = q(\zeta )\) is defined by Eq. (13). Using the coarea formula gives

$$\begin{aligned} I(q) =\int _0^1\int _{\varSigma (s)} \frac{\mathrm {e}^{-\beta F(\zeta )}\nabla q(\zeta )^{\mathsf {T}}G(\zeta )^{-1}\nabla q(\zeta )}{|\nabla \sigma (\zeta )|} \mathrm {d}S_\zeta \,\mathrm {d}s. \end{aligned}$$

Using \(\nabla q(\zeta ) = {\hat{q}}_s(s)\nabla \sigma (\zeta )\) gives

$$\begin{aligned} I(q)= & {} \int _0^1{\hat{q}}_s(s)^2 \int _{\varSigma (s)}\mathrm {e}^{-\beta F(\zeta )} \nabla \sigma (\zeta )^{\mathsf {T}}G(\zeta )^{-1}\hat{n}(\zeta ) {\mathrm {d}}S_\zeta \,{\mathrm {d}}s \\= & {} \int _0^1{\hat{q}}_s(s)^2 Q(s){\mathrm {d}}s \end{aligned}$$

where \(\hat{n}(\zeta ) = \nabla \sigma (\zeta )/|\nabla \sigma (\zeta )|\) is the outer normal at \(\zeta\) on \(\varSigma (s)\) and Q(s) is given by Eq. (15). Minimizing over \({\hat{q}}\) for fixed \(\sigma\) gives the Euler–Lagrange equation \(-(2{\hat{q}}_s Q(s))_s = 0\), whose solution is given by

$$\begin{aligned} q(Z(s)) = {\hat{q}}(s) = I(q)\int _0^s\frac{\mathrm {d}s}{Q(s)}. \end{aligned}$$


$$\begin{aligned} I(q) = \left( \int _0^1 \frac{\mathrm {d}s}{Q(s)}\right) ^{-1}. \end{aligned}$$

From Eqs. (33) and (32), one obtains Eq. (14).

1.4 Direction of steepest descent

Let \((\int _0^1 \delta Z(s)^{\mathsf {T}}G(Z(s))\delta Z(s){\mathrm {d}}s)^{1/2}\) measure the size of a deviation \(\delta Z(s)\) from a path Z(s). For the direction of steepest descent f for the functional (17) subject to the constraint (23) the Lagrangian is

$$\begin{aligned} \int _0^1 f^{\mathsf {T}}h\,{\mathrm {d}}s +\mu \left( \int _0^1 f^{\mathsf {T}}G f\,\mathrm {d}s -{\mathrm {const}}\right) +\int _0^1\lambda ((f^{\mathsf {T}}b)_s + (f^{\mathsf {T}}\nabla _\omega c)_{ss}){\mathrm {d}}s. \end{aligned}$$

Its first variation is

$$\begin{aligned} \int _0^1 (\delta f)^{\mathsf {T}}(h -\lambda _s b +\lambda _{ss}\nabla _\omega c + 2\mu G f){\mathrm {d}}s + \left[ (\delta f)_s^{\mathsf {T}}\lambda \nabla _\omega c\right] _0^1. \end{aligned}$$

The Euler–Lagrange equations give \(\lambda (0) =\lambda (1) = 0\) and \(f = -(2\mu G)^{-1}(h -\lambda _s b +\lambda _{ss} \nabla _\omega c)\), in addition to the two original constraints. The value of \({\mathrm {const}}\) can be chosen to make \(\mu =\frac{1}{2}\), giving Eq. (25). With this choice, Eq. (23) becomes

$$\begin{aligned} (-\lambda _{sss} + b^{\mathsf {T}}G^{-1} b\lambda _s - b^{\mathsf {T}}G^{-1}h)_s = 0. \end{aligned}$$

where \(b^{\mathsf {T}}G^{-1}\nabla _\omega c = 0\) follows from Eq. (20). Also, using Eq. (20), one has for \(s = 0, 1\) that

$$\begin{aligned} 0 = Z_s^{\mathsf {T}}G f = -\lambda _{ss}Z_s^{\mathsf {T}}\nabla _\omega c =\lambda _{ss}c^2, \end{aligned}$$

thus obtaining two more boundary conditions for \(\lambda\), namely \(\lambda _{ss}(0) =\lambda _{ss}(1) = 0\).

1.5 Most probable path

As a first step in maximizing the action, given by Eq. (26), separate the path of the trajectory from its kinetics by a reparameterization \(t =\tau (s)\), \(0\le s\le 1\), and define the path by

$$\begin{aligned} Z(s) ={\hat{\zeta }}(\tau (s)). \end{aligned}$$

(For example, choose \(\tau (s)\) so that s is relative arclength, meaning that \(|Z_s(s)| = L\) where L is the length of the path.) With this change of variables, the action becomes

$$\begin{aligned}S[{\hat{\zeta }}] =\int _0^1\left| \tau _s^{-1}Z_s + f(Z)\right| _Z^2\tau _s\,{\mathrm {d}}s. \end{aligned}$$

The schedule \(t =\tau (s)\) that minimizes the action subject to \(\tau (0) = 0\) and \(\tau (1) = T\) is determined using the calculation of variations. The result is that \(\tau _s = |Z_s|_Z(|f|_Z^2 + C)^{-1/2}\) where C satisfies Eq. (28). With this choice, the action becomes \({\bar{S}}[Z]\), given in Eq. (27).

Consider now the effect of a change of variables on the functional. In particular, if one use instead variables \(\eta\), where

$$\begin{aligned} \zeta\,=\,\chi (\eta ), \end{aligned}$$

one has \(Z = \chi (\bar{Z})\) and \(\bar{F}(\eta )\), \(\bar{D}(\eta )\) given by

$$\begin{aligned} \bar{F}= & {} F(\chi ) -\frac{1}{\beta }\log \left| \det \frac{\partial \chi }{\partial \eta }\right| , \\ \bar{D}= & {} \left( \frac{\partial \chi }{\partial \eta }\right) ^{-1} D(\chi )\left( \frac{\partial \chi }{\partial \eta }\right) ^{-{\mathsf {T}}}, \end{aligned}$$

where the last two equations are derived from definitions given by Eqs. (2) and (3). Therefore, \(\bar{Z}_s = ((\partial \chi /\partial \eta )(\bar{Z}))^{-1}Z_s\) and \(\bar{s}[\bar{Z}] = \bar{S}[Z]\) provided \(\bar{f} =(\partial \chi /\partial \eta )^{-1}f(\chi )\) where

$$\begin{aligned} \bar{f} =\beta \bar{D}\nabla _\eta \bar{F} -{\mathrm {div}}_\eta \bar{D}. \end{aligned}$$

Differentiating the expression for \(\bar{F}\) gives

$$\begin{aligned} \left( \frac{\partial \chi }{\partial \eta }\right) ^{\mathsf {T}}(\nabla _\zeta F)(\chi ) =\nabla _\eta \left( \bar{F} +\frac{1}{\beta }\log \left| \det \frac{\partial \chi }{\partial \eta }\right| \right) , \end{aligned}$$


$$\begin{aligned} f(\chi ) =\beta D(\chi )(\nabla _\zeta F)(\chi ) -({\mathrm {div}}D)(\chi ) = \frac{\partial \chi }{\partial \eta }\bar{f} + g, \end{aligned}$$


$$\begin{aligned} g =\frac{\partial \chi }{\partial \eta }\bar{D}\nabla _\eta \left( \log \left| \det \frac{\partial \chi }{\partial \eta }\right| \right) -({\mathrm {div}}D)(\chi ) +\frac{\partial \chi }{\partial \eta }{\mathrm {div}}\bar{D}. \end{aligned}$$

One has

$$\begin{aligned} \left( \frac{\partial \chi }{\partial \eta }\bar{D}\nabla \left( \log \left| \det \frac{\partial \chi }{\partial \eta }\right| \right) \right) ^i =\chi ^i_j\bar{D}^{jk}\left( \log \left| \det \frac{\partial \chi }{\partial \eta }\right| \right) _k =\chi ^i_j\bar{D}^{jk} {\mathrm {tr}}\left( \left( \frac{\partial \chi }{\partial \eta }\right) ^{-1}\left( \frac{\partial \chi }{\partial \eta }\right) _k\right) , \end{aligned}$$

where tensor notation is used, with subscripts for differentiation. Differentiating \(D^{ij}(\chi )\) gives

$$\begin{aligned} D^{ij}_k(\chi )\chi ^k_l = (\chi ^i_m\bar{D}^{mn}\chi _n^j)_l, \end{aligned}$$


$$\begin{aligned} D^{ij}_k(\chi ) = (\chi ^i_m\bar{D}^{mn}\chi _n^j)_l\bar{\chi }^l_k, \end{aligned}$$

where \(\bar{\chi }^i_j\) is element ij of \((\partial \chi /\partial \eta )^{-1}\), so

$$\begin{aligned} ({\mathrm {div}}D)^i(\chi )=\, (\chi ^i_m\bar{D}^{mn}\chi ^j_n)_l\bar{\chi }^l_j \\=\, \chi ^i_{ml}\bar{D}^{ml} +\chi ^i_m\bar{D}^{ml}_l +\chi ^i_m\bar{D}^{mn}\chi ^j_{nl}\bar{\chi }^k_j. \end{aligned}$$


$$\begin{aligned} g^i ={\mathrm {tr}}\left( \bar{D}\frac{\partial ^2\chi ^i}{\partial \eta \partial \eta } \right) . \end{aligned}$$

The extra term g changes the form of the functional to be minimized, so if one minimizes the integral using instead variables \(\eta\), the resulting path \(\eta = \bar{Z}(s)\) does not satisfy \(Z(s) = \chi (\bar{Z}(s))\).

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Skeel, R.D., Zhao, R. & Post, C.B. A minimization principle for transition paths of maximum flux for collective variables. Theor Chem Acc 136, 14 (2017).

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