Nonequilibrium Markov state modeling of the globule-stretch transition

We describe a systematic approach to construct coarse-grained Markov state models from molecular dynamics data of systems driven into a nonequilibrium steady state. We apply this method to study the globule-stretch transition of a single tethered model polymer in shear flow. The folding and unfolding rates of the coarse-grained model agree with the original detailed model. We demonstrate that the folding and unfolding proceeds through the same narrow region of configuration space but along different cycles.

Figure 1

Mean end-to-end distance $\langle R_{\text{ee}}\rangle$ as a function of strain rate $\dot{\gamma }$. For $\dot{\gamma }=1.6$ [yellow (bright) circle] the inset shows an exemplary time series. Also shown are two snapshots for the globule (bottom) and extended polymer (top).

Figure 2

Configuration space ($\dot{\gamma }=1.6$). Scatter plot of the normalized projections of all centroids onto the two largest principal components. Colors indicate the end-to-end distance $R_{\text{ee}}$.

Figure 3

Cycle space ($\dot{\gamma }=1.6$). (a) Scatter plot of cycle center vs. the cycle diameter, and (b) cycle affinity vs. cycle center. The cycle diameter and centers are computed using a PCA (see text for details). Colors represent cycle communities while gray points indicate no community. The filled black circles indicate the cycle representatives. [(c) and (d)] The same plots as for (a) and (b) but for three cycle communities. (e) The fuzzy partition coefficient computed for multiple communities. The best result is obtained for three communities followed by a slightly lower value for five communities.

Figure 4

Final NE-MSM for $\dot{\gamma }=1.6$. (a) Transition network of the polymer dynamics with 15 centroids (filled circles) and five cycles. The colored states green, blue, and red correspond to the colors of the local cycle communities shown in Fig. 3. States with a black border are structurally very similar and constitute the transition ensemble. The arrows point in the direction of probability currents (net flux), while the arrow widths represent the magnitude of currents. On average, the polymer dynamics follows the direction of the arrows. Arrow colors cyan and magenta correspond to the global cycles. (b) Transition network in configuration space using the normalized projections onto the two largest principal components. Symbols are the configurations while lines indicate the cycles with the arrows pointing in the direction of the net flux.

Figure 5

Lag time analysis. The six slowest internal time scales are plotted for different lag times. For building the MSM we choose $\tau =20$.

Figure 6

Illustration of bridge and triangle state coarse-graining approach. (a) Bridge state: After deletion of state 1 the other states are directly connected (dotted arrow). (b) Triangle state: After deletion of state 1 the existing connections $0\leftrightarrow 2$ are modified. The arrows point in the direction of probability currents.