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Chinese Physics B
Vol. 29, Issue 10, (2020)

Find slow dynamic modes via analyzing molecular dynamics simulation trajectories

Chuanbiao Zhang¹ and Xin Zhou^2,†

Author Affiliations

¹College of Physics and Electronic Engineering, Heze University, Heze 27405, China

²School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

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DOI: 10.1088/1674-1056/abad24 Cite this Article

Chuanbiao Zhang, Xin Zhou. Find slow dynamic modes via analyzing molecular dynamics simulation trajectories[J]. Chinese Physics B, 2020, 29(10): Copy Citation Text

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Abstract

It is a central issue to find the slow dynamic modes of biological macromolecules via analyzing the large-scale data of molecular dynamics simulation (MD). While the MD data are high-dimensional time-successive series involving all-atomic details and sub-picosecond time resolution, a few collective variables which characterizing the motions in longer than nanoseconds are needed to be chosen for an intuitive understanding of the dynamics of the system. The trajectory map (TM) was presented in our previous works to provide an efficient method to find the low-dimensional slow dynamic collective-motion modes from high-dimensional time series. In this paper, we present a more straight understanding about the principle of TM via the slow-mode linear space of the conformational probability distribution functions of MD trajectories and more clearly discuss the relation between the TM and the current other similar methods in finding slow modes.

Keywords

molecular dynamics simulation slow modes trajectory map

\begin{array}{r} γ d \boldsymbol q = - \frac{\partial U}{\partial \boldsymbol q} d t + \sqrt{2 k_{B} T γ d t} d \boldsymbol G, \end{array}

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\begin{array}{r} \frac{\partial}{\partial t} P (q, t) = {\boldsymbol L}_{F P} P (q, t), \end{array}

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(1)

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\begin{array}{r} - \frac{\partial}{\partial t} Ψ (q, t) = \boldsymbol H Ψ (q, t), \end{array}

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(1)

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\begin{array}{r} P (q, t) = P_{e q} (q) \sum_{n = 0}^{\infty} a_{n} (0) e^{- λ_{n} t} {\hat{Φ}}_{n} (q), \end{array}

(4)

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\begin{array}{r} \frac{P (q, t)}{P_{e q} (q)} \approx 1 + \sum_{n = 1}^{N} a_{n} (t) {\hat{Φ}}_{n} (q), \end{array}

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\begin{array}{r} {\hat{Φ}}_{n} (q) = \frac{\sum_{i} b_{n, i} P_{i} (q)}{P_{e q} (q)}, \end{array}

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\begin{array}{r} {\hat{Φ}}_{n} (q) = \sum_{α} c_{n α} Θ_{α} (q) . \end{array}

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\begin{array}{r} \frac{P_{i} (q)}{P_{e q} (q)} = \sum_{α} c_{i α}^{^{'}} Θ_{α} (q), \end{array}

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\begin{array}{r} \frac{P_{i} (q)}{P_{e q} (q)} \approx \sum_{μ} p_{i, μ} A^{μ} (q) . \end{array}

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\begin{array}{r} {⟨ A_{μ} (q) A^{ν} (q) ⟩}_{e q} = δ_{μ ν}, \end{array}

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\begin{array}{r} {\boldsymbol p}_{i} = p_{i, μ} {\hat{A}}^{μ} (q), \end{array}

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\begin{array}{r} {\hat{B}}_{α} (q) = b_{α μ} {\hat{A}}^{μ} (q), \end{array}

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\begin{array}{r} {\hat{B}}_{α} (t) = \frac{1}{Δ t} \int_{t}^{t + Δ t} {\hat{B}}_{α} (q (t^{^{'}})) d t^{^{'}} . \end{array}

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\begin{array}{r} G (q, t | q_{0}, 0) = P_{e q} (q) [1 + \sum_{n = 1}^{N} e^{- λ_{n} t} {\hat{Φ}}_{n} (q) {\hat{Φ}}_{n} (q_{0})], \end{array}

(14)

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\begin{array}{r} \begin{array}{lll} G_{ν}^{μ} (t, 0) & = & \int d q d q^{^{'}} A^{μ} (q) G (q, t | q^{^{'}}, 0) A_{ν} (q^{^{'}}) P (q^{^{'}}, 0) \\ \equiv & {⟨ A^{μ} (q (t)) A_{ν} (q (0)) ⟩}_{0}, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} G_{ν}^{μ} (t) & = & \frac{1}{{\hat{Z}}_{μ}} {⟨ θ_{μ} (q (t)) θ_{ν} (q (0)) ⟩}_{e q} \\ = & \frac{n_{μ ν} (t, 0)}{n_{ν} (0)}, \end{array} \end{array}

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\begin{array}{r} Σ^{μ ν} = \frac{1}{τ} \int_{0}^{τ} d t (1 - \frac{t}{τ}) [C_{μ ν} (t) + C_{ν μ} (t)], \end{array}

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\begin{array}{r} C_{μ ν} (t) = \frac{1}{τ - t} \int_{0}^{τ - t} d t_{1} G_{μ ν} (t + t_{1}, t_{1}) . \end{array}

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Chuanbiao Zhang, Xin Zhou. Find slow dynamic modes via analyzing molecular dynamics simulation trajectories[J]. Chinese Physics B, 2020, 29(10):

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