Coarse graining of biochemical systems described by discrete stochastic dynamics

Many biological systems can be described by finite Markov models. A general method for simplifying master equations is presented that is based on merging adjacent states. The approach preserves the steady-state probability distribution and all steady-state fluxes except the one between the merged states. Different levels of coarse graining of the underlying microscopic dynamics can be obtained by iteration, with the result being independent of the order in which states are merged. A criterion for the optimal level of coarse graining or resolution of the process is proposed via a tradeoff between the simplicity of the coarse-grained model and the information loss relative to the original model. As a case study, the method is applied to the cycle kinetics of the molecular motor kinesin.

Figure 1

A graph with one arbitrarily chosen spanning tree. The edges missing in the spanning tree are called chords and define fundamental cycles. The blue dotted edge $1\to 2$ is a chord and defines the first (fundamental) cycle, while the red dashed chord $3\to 4$ defines the second (fundamental) cycle. The third cycle can be constructed by fundamental cycles [1]. The choice of a spanning tree of a graph is arbitrary. The reference orientation of all cycles is chosen to be counterclockwise.

Figure 2

State $m$ and state $n$ will be merged such that the fluxes between the merged state and the rest of the system are retained and the probabilities add up.

Figure 3

Kinetic diagram for kinesin from Ref. [18]. The six-state model will be simplified to a four-state model without changing the number of cycles within the network. In the forward cycle, states 1 and 6 will be merged to state 1”, while in the backward cycle, states 3 and 4 are described by state 3” in the coarse-grained network. Except for the fluxes between the merged states, all net transition fluxes are retained. We eliminated the transition with minimal contribution to entropy production in each fundamental cycle.

Figure 4

Simulation results for the velocity (a) and the entropy production (b) of a kinesin motor. The rate constants for the six-state model are from Ref. [18] for chemical concentrations $\left[ATP\right]=\left[ADP\right]=\left[P\right]=1\mu M$, stepping size $l=8\mathrm{nm}$ and an external load force $F=1\mathrm{pN}$. 10 000 trajectories have been sampled for each model with a simulation time $\tau =1200\mathrm{s}$. The black data (circles) corresponds to the original six-state model from Ref. [18], which is depicted in Fig. 3. The blue data (squares) represents a coarse-grained system with five states, where the original states 3 and 4 have been merged. The four-state model is depicted in red and with triangles. The entropy production for a trajectory of finite length was calculated as in Eq, (42). For the entropy production, the four-state model underestimates the mean by around 17%. The velocity $v$ of a kinesin motor is proportional to the step size $l=8$ nm and to the flux along the mechanical transition ($J_{25}$ in the six-state, $J_{2^{\prime }{4}^{\prime }}$ in the six-state, and $J_{2^{\prime \prime }{4}^{\prime \prime }}$ in the four-state model). Our coarse-graining approach preserves the mean of the velocity whereas it does not preserve variances of observables. However, the numerical differences for the variance of the velocity are very small.

Figure 5

Different models for a kinesin motor. In (a), no external stall force is applied and the forward cycle is dominant. In (b), the kinesin motor is at an external load force of 6.5 pN, which is a substantial force, but smaller than the stall force. In (c), the applied load force is larger than the stall force and the backward cycle is dominant.

Figure 6

In (a) and (d), the mean velocity and entropy production are plotted against the external load force acting on the motor for different kinesin models, see Fig. 5. The figures in the second column, (b) and (e), show an enlarged view of the differences between tricyclic and uni-cyclic systems for large external load forces. Furthermore, the relative differences between the mean velocity and the entropy production for coarse-grained models and the original six-state model, respectively, are depicted in the third column in (c) and (f). The tricyclic four-state model is optimal in terms of preserving the velocity of the motor. Moreover, the load force ranges indicated in (f) show which model is most suitable for approximating the entropy production within the given range. The motor has zero mean velocity for $F=7\mathrm{pN}$. For no or small external load forces, unicyclic models resembling the forward cycle approximate the mean velocity and entropy production well with small deviations. For load forces around the stall force, on the other hand, a coarse-grained model is better provided it has the same network topology as the original model. In the Appendix, the variances of velocity and entropy production are plotted against the external load force.

Figure 7

In the first row, the blue arrows contribute to the forward cycle $F$ in the positive ($+$) direction (counterclockwise). Its algebraic value is ${\pi }_{F+}={\alpha }_{12}{\alpha }_{25}{\alpha }_{56}{\alpha }_{61}$. The dashed arrows flow into the forward cycle $F$. They correspond to the factor $S_C$ in Eq. (A5). In the second row, the diagrams for the forward cycle in plus direction are depicted for a coarse-grained network as shown in Fig. 3. States 1 and 6 in the original network are merged to state 1' in the coarse grained network. States 3 and 4 are merged to state 3'.

Figure 8

The variance of the velocity and the entropy production are plotted against the external load force acting on the motor for different kinesin models. Furthermore, the relative differences between the variance of the velocity and the entropy production for coarse-grained models and the original six-state model, respectively, are depicted in the second column. The relative differences in the second column reveal large approximation errors by unicyclic systems that resembles the forward cycle for large external load forces. The motor has zero mean velocity for $F=7\mathrm{pN}$. For no or small external load forces, unicyclic models approximate the variances for the velocity and the entropy production well with small deviations while, for larger load forces, a coarse-grained model is better, if it has the same network topology as the original model. Both quantities are calculated analytically as described in Sec. 3. The variance was calculated for trajectories with length $\tau =1\mathrm{s}$.