Universal bounds on entropy production inferred from observed statistics

Nonequilibrium processes break time-reversal symmetry and generate entropy. Living systems are driven out-of-equilibrium at the microscopic level of molecular motors that exploit chemical potential gradients to transduce free energy to mechanical work, while dissipating energy. The amount of energy dissipation, or the entropy production rate (EPR), sets thermodynamic constraints on cellular processes. Practically, calculating the total EPR in experimental systems is challenging due to the limited spatiotemporal resolution and the lack of complete information on every degree of freedom. Here, we propose a new inference approach for a tight lower bound on the total EPR given partial information, based on an optimization scheme that uses the observed transitions and waiting times statistics. We introduce hierarchical bounds relying on the first- and second-order transitions, and the moments of the observed waiting time distributions, and apply our approach to two generic systems of a hidden network and a molecular motor, with lumped states. Finally, we show that a lower bound on the total EPR can be obtained even when assuming a simpler network topology of the full system.

Living systems operate far-from-equilibrium and constantly produce entropy.At the molecular level, the hydrolysis of fuel molecules, such as Adenosine triphosphate (ATP), powers nonequilibrium cellular processes, utilizing part of the liberated free energy for physical work, while the rest is dissipated [5].The dissipation, or entropy production, is a signature of irreversible processes and can be used as a direct measure of the deviation from thermal equilibrium [19][20][21][22].Therefore, the entropy production rate plays an important role in our understanding of the physics, and underlying mechanism, governing biological and chemical processes [13-15, 17, 18, 23].
Estimating the total EPR is only possible if we have knowledge regarding all of the degrees of freedom that are out-of-equilibrium [44].However, due to practical limitations on the spatiotemporal resolution, not all of them can be experimentally accessible, and one can only obtain a lower bound on the total EPR for partially observed or coarse-grained systems [45].
The passive partial entropy production rate, σ pp , is an estimator for the EPR calculated from the transitions between two observed states, which bounds the total EPR [45][46][47][48].This estimator, however, fails to provide a nonzero bound in case of vanishing current over the observed link, i.e., at stalling conditions [45].Other EPR estimators for partially observed systems based on inequality relations like the TUR [24-26, 32, 49] also fail to provide a non-trivial bound on the total EPR in the absence of net flux in the system.
The Kullback-Leibler Divergence (KLD) estimator, σ KLD , is based on the KLD, or the relative entropy, between the time-forward and the time-revered path probabilities [21,[50][51][52][53][54][55].For semi-Markov processes, this estimator is a sum of two contributions.The first stems from transitions irreversibility or cycle affinities, σ aff , whereas the second stems from broken time-reversal symmetry reflected in irreversibility in waiting time distributions (WTD), σ WTD [56].Using the KLD estimator, one can obtain a non-trivial lower bound on the total EPR for second-order semi-Markov processes even in the absence of the net current [35,[56][57][58][59].Moreover, a lower bound on the total EPR can be obtained from the KLD between transition-based WTD [52,57,60].
Recently developed estimators solved an optimization problem to obtain a lower bound on the entropy production.For a discrete-time model, Ehrich proposed to search over the possible underlying systems that maintain the same observed statistics using knowledge on the number of hidden states [61].For continuous-time models, Skinner and Dunkel minimized the EPR on a canonical form of the system that preserved the first-and secondorder transition statistics to yield a lower bound on the total EPR, σ 2 [62].The authors also formulated an optimization problem to infer the EPR in a system with two observed states using the waiting time statistics [34].
In this paper, we provide a tight bound on the total EPR by formulating an optimization problem based on the statistics of both transitions and waiting times.We use the first-and second-order statistics for the mass transition rates, and any chosen number of moments of the observed waiting time distributions.For a system with a known topology, we calculate the analytical expressions of the statistics as functions of the mass rates and the steady-state probabilities, which describe a possible underlying system and are used as variables in the optimization problem.These analytical expressions are then used to constrain the optimization variables to match the observed statistics.We show for a few continuous-time Markov chain systems that using the constraints of the mass rates and only the first moment of the WTD already provides close-to-total EPR value.Our approach outperforms other estimators, such as σ pp , σ KLD , σ aff , and σ 2 , in terms of the tightness of the lower bound.In the case of a complex model, where the formulation of the optimization problem might not be practical due to the number of constraints, or in case the full topology is not known, we show numerically that assuming a simpler underlying topology can provide a lower bound on the total EPR.
The paper is organized as follows.In section II, we describe our model system and the coarse-graining approach.The results are presented in section III: We discuss the estimator in subsection III A, apply it to different systems in subsection III B, demonstrate how the accuracy of the measured statistics affects the results of our estimator in subsection III C, and finally, we show the results of the optimization problem assuming a simpler underlying model in subsection III D. We conclude our findings in section IV.

II. MODEL
We assume a continuous time Markov chain over a finite and discrete set of states i = {1, 2, . . ., N }.A trajectory is described by a sequence of states and their corresponding residence times before a transition to the next state occurs.Being a Markovian process, the jump probabilities depend only on the current state.
The transition rates w ij from state i to j determine the time evolution of the probabilities for the system to be in each state, according to the Master equation d dt p(t) T = p(t) T W , where T is the transpose operator, and W is the rate matrix p(t) is a column vector of the state probabilities at time t, with i p i (t) = 1, and the diagonal entries are calculated according to λ i = j =i w ij for probability conservation.At the long-time limit, the system eventually reaches a steady state π, where lim t→∞ p i (t) = π i such that 0 = The waiting time at each state i is an exponential random variable with mean waiting time of τ i = λ −1 i .The mass rates n ij are defined as follows: The probabilities of jumping from state i to state j can be written in terms of the mass transition rates: The steady-state total EPR can be calculated by multiplying the net currents and the mass rate ratios (affinities), summing over all the links [5,6]: Given a long trajectory of a total duration T , the steadystate probability π i is the fraction of time spent in state i, and the mass rate n ij is the number of transitions i → j divided by T .
according to the definition of the mass transition rates in Eq. 2, at the steady state, a mass conservation is satisfied at each state: In many practical scenarios, some of the microstates cannot be distinguished, and the transitions between them cannot be observed.In such a case, a set of states {i 1 , i 2 , . . ., i N I } is observed as a single coarse-grained state I (Fig. 1(a)).The observed trajectory, therefore, includes only coarse-grained states and the combined residence time (Fig. 1(b)), and it is not necessarily a Markovian process [56].Such a decimation procedure of lumping several states can give rise to semi-Markovian processes of any order depending on the topology of the network [62,[64][65][66].In this case, the observed statistics of two or more consecutive transitions may give us additional information on the process.

A. Bounding the entropy production rate
Given a coarse-grained system with a model of the full underlying Markovian network topology, we can formulate an optimization problem for obtaining a tight bound on the total EPR.We consider a few observables: the coarse-grained steady-state probabilities, π I , which is the probability to observe the system in the coarse-grained state I; the first-order mass transition rates, n IJ , which is the rate of observing the transition I → J; the second order mass transition rates, n IJK , which is the rate of observing the transition I → J followed by the transition J → K; and the conditional waiting time distributions ψ IJK (t), which is the distribution of waiting times in a coarse-grained state J before a transition to a coarsegrained state K occurs, conditioned on the previous transition being I → J.
We search over the space of all possible underlying systems with the same topology as our hypothesized Markovian model that give rise to the same observed statistics, while minimizing the EPR.Trivially, the EPR of the coarse-grained system at hand is bounded from below by the EPR of the underlying Markovian system with the same observed statistics after coarse-graining, having the minimal value of entropy production.

Analytical expressions of the observed statistics
The observed statistics of the coarse-grained system can be expressed analytically in terms of the mass rates and steady-state probabilities of the model underlying system.From probability and mass conservation, π I = i∈I π i , and n IJ = i∈I,j∈J n ij , respectively.The mass conservation for the second-order transitions n IJK must include all the paths starting at state i ∈ I, passing through a state in J, where any number of transitions might occur inside J, and jumping to state k ∈ K. To account for the transitions within J, we define the matrix P JJ of the transition probabilities between states in J, j m , j n ∈ J: Summing over the possible transitions from I, transitions within J, and transitions to K, we have (see Appendix A): where I is the identity matrix of the size of P JJ , and n iJ and p Jk are column vectors of the mass transition rates from state i ∈ I to any state j ∈ J, and jump probabilities from any state j ∈ J to a state k ∈ K, respectively: and: The conditional waiting time distribution ψ IJK (t) can be calculated by the Laplace and inverse-Laplace transforms (full derivations can be found in Appendix B).We start from the Laplace transform of ψ ij (t) = w ij e −λit , the joint probability distribution of the transition i → j and the waiting time in the Markovian state i: Note that for any function . Now, we consider the simple case where the secondorder transition through the coarse-grained state J starts and ends in specific Markovian states i ∈ I and k ∈ K, respectively.The Laplace transform of the distribution of waiting times in J before a transition to k occur, given the previous transition was i → J is: where and ΨJJ (s) is a matrix of the Laplace transforms of every joint probability distribution of waiting times and transitions within J: We denote ψiJK (s) ≡ k∈K ψiJk (s).Then, the Laplace transform of the conditional waiting time distribution is: Finally, we apply an inverse Laplace transform to obtain the conditional probability density: We further impose mass conservation at each of the Markovian states according to Eq. 5, to make sure the solution represents a valid Markovian system.

Formalizing the optimization problem
Let S be the real underlying Markovian system and let R be a general underlying system with the same topology as S, i.e., the same states and possible transitions as S, but R can have arbitrary mass rates and steadystate probabilities.Given the set of all systems R with the same steady-state probabilities π R I = π S I , same firstorder mass transition rates n R IJ = n S IJ , same second-order mass transition rates n R IJK = n S IJK , and the same conditional waiting time distributions ψ R IJK (t) = ψ S IJK (t), as the system S, the following inequality holds for the EPR of S and R, σ(S) and σ(R), respectively: where σ opt is the minimal EPR value of all the possible underlying systems R. The inequality holds since the real system S belongs to the set of systems over which we minimize.The only variables of the optimization problem are n ij and π i , from which one can fully describe any of the possible underlying Markovian systems R. All the constraints, π I , n IJ , n IJK , and ψ IJK (t), as well as the EPR objective function, depend on these variables.Note that these variables are bounded by 0 In contrast to the constraints on the steady-state probabilities and the first-and second-order mass transition rate values, the constraint on the waiting-time distributions requires an equality of continuous functions ψ IJK (t), which one cannot fully reconstruct from trajectory data of finite duration.Moreover, solving the optimization problem using a constraint on a function with non-trivial dependency on the optimization problem variables is extremely challenging.Thus, we modify the optimization, and instead, use the moments of the waiting time distributions: where t k IJK is the k-th moment of the conditional waiting time distribution ψ IJK (t).Using increasing number of moments, we can write the hierarchical bounds: We can easily get the analytical expressions for the moments t k IJK from the Laplace transform (see Appendix B): Now, for each moment, we have an expression that depends on the optimization problem variables in a simpler way, which in turn, simplifies the calculations.After calculating the values of the observables for the optimization problem, we solve it using a global search non-linear optimization algorithm [67].

4-state system
We consider a fully-connected network of 4 states, with two observed states {1, 2} and two hidden states {3, 4}, which are coarse-grained to state H (Fig. 2(a)), resulting in second-order semi-Markov dynamics [56].The observed statistics of interest are the steady state probabilities π 1 , π 2 and π H , the first-order mass transition rates n 1H , n H1 , n 2H , and n H2 , the second-order mass transition rates n 1H2 and n 2H1 and the k-th moment of the conditional waiting time distributions t k 1H1 , t k 1H2 , t k 2H1 and t k 2H2 .Notice we only used the second-order statistics through the coarse-grained state H, since states 1 and 2 are Markovian.Furthermore, we do not use n 1H1 and n 2H2 since they depend on the other mass transition rates: The derivations of the analytical expressions of the secondorder mass transition rates and the moments of the conditional waiting time moments, for this system, can be found in Appendix C.
We tune the transition rates over the observed link between states 1 and 2 according to w 12 (F ) = w 12 e −βF L and w 21 (F ) = w 21 e βF L , where β = T −1 is the inverse temperature (with k B = 1), and L is a characteristic length scale, to mimic external forcing.We compare the different EPR estimators on the system for several values for a driving force F over the observed link (Fig. 2(b)).
The passive partial EPR [45]: The KLD estimator is the sum of two contributions: where p([IJ] → [JK]) is the probability to observe the transition J → K given the previous transition was I → J, p IJK is the probability to observe the secondorder transition I → J → K, and D[p||q] is the KLD between the probability distributions p and q.As was previously shown, the hierarchy between the EPR estimators is σ KLD ≥ σ aff ≥ σ pp [45,56].The σ 2 estimator is also formulated as an optimization problem searching over a canonical form of the system with the same observed statistics, however, it only considers the first-and second-order mass transition rates [62].Its place in the hierarchy between the EPR estimators varies for different systems.While σ 2 can be greater opt (brown cross), KLD estimator σKLD (dotted blue line), affinity estimator σ aff (dashed green line), two-step estimator σ2 (yellow Asterisk), and the passive partial entropy production σpp (dashed-dotted orange line).The rates we used are than σ KLD in some cases [62], here, for the rate values we used, σ 2 < σ KLD .In fact, although the values of σ 2 and σ aff appear to be similar (Fig. 2(b)), actually σ 2 < σ aff for all of the values of F used.
At the stalling force, there is no current in the visible link and we get σ pp = σ aff = σ 2 = 0, which is the trivial bound.In contrast, σ KLD and our estimator σ opt surpasses σ KLD significantly and yields a tight bound.For this system, using higher moments in order to calculate σ (2) opt did not make any improvement compared to σ (1) opt .

Molecular motor
Here, we study a model of a molecular motor, illustrated in Fig. 3(a).The motor can physically move in space (upward or downward), i ↔ i + 1, or change internal states (passive or active), i ↔ i .An external source of chemical work ∆µ drives the upward spatial jumps from the active state, and a mechanical force F acts against it and drives the downward transitions.We assume that an external observer cannot distinguish between the internal states of the motor, but rather can only record its physical position.The observed statistics are thus of a second-order Semi-Markov process [56].
Owing to the transnational symmetry in the model, we represent the molecule motor as a cyclic network of three coarse-grained states where each of them represents the physical location, lumping the active and passive internal states.We denote the steady-state probability of being in the passive and active states as π and π , respectively.Notice that the probability to be in each physical location in the 3-state cyclic system is the same, and that π and π are the same for all of the physical locations, therefore, π + π = 1/3.
We denote the upward and downward transitions from and to the passive state as u 1 and d 1 , respectively, the upward and downward transitions from and to the active state as u 2 and d 2 , respectively, and the transitions between the active and passive states at the same physical location as r (right) and l (left), respectively.The upward and downward coarse-grained transitions are labeled as U and D, respectively.
The observed statistics of interest are the first-order mass rates n U , n D , the second-order mass rates n U U , n DD and the k-th moment of the conditional waiting times t k U U , t k U D , t k DU and t k DD .Note that we do not use n U D and n DU , since they depend on the other mass rates: n U D = n U −n U U and n DU = n D −n DD .Owing to the symmetry of the cycle representation of the coarse-grained system, in which the steady-state probabilities are equally distributed, we only need the constraints on the upward and downward transitions.The derivations of the analytical expressions of the secondorder mass transition rates and the moments of the conditional waiting time distributions, for this system, can be found in Appendix D.
The chemical affinity µ, arising from ATP hydrolysis for example, only affects the transitions u 2 and d 2 , whereas the external force F affects all of the spatial transitions u 1 , d 1 , u 2 and d 2 .The transition rates then obey local detailed balance: w d1 /w u1 = e βF L and w d2 /w u2 = e β(F L−µ) , where L is the length of a single spatial jump [56].
We compare the different EPR estimators for the molecular motor system for several values of µ and for each µ value, we tune the external forcing parameter F (Fig. 3(b)).Notice the passive partial EPR, σ pp , is not applicable for this system since all the original Markovian states are coarse-grained.
The hierarchy of the different EPR estimators for the molecular motor, for the rate values we used, is σ At the stalling force for each value of µ, where there is no visible current, we find σ aff = σ 2 = 0, which is the trivial bound.In contrast, similar to the 4state system, σ opt surpasses σ KLD significantly and yields a tight bound.

C. Importance of data accuracy
One of the hyper parameters defining the optimization problem is the constraint tolerance, which indicates the acceptable numerical error of the solution.If is the absolute error of the trajectory statistics with respect to the true analytical ones, then the constraint tolerance must be equal to or greater than .Otherwise, the optimiza- tion problem might not converge or give an overestimate in the worst-case scenario.
In Fig. 4, we plot the absolute (and relative) error of a few statistics values calculated from several trajectories as function of the trajectory length N , for both systems discussed in the previous sections.Moreover, using the analytical values of the statistics for maximum accuracy, we plot the results of our estimator σ opt assuming the simple molecular motor model (2 hidden states), when the real system has 2 (red cross), 3 (green triangles) or 4 (blue circle) hidden states.For both systems, the results are presented for random generated transition rates (for each case) with statistics calculated from trajectories of length N = 10 8 using a constraint tolerance of 10 −5 .
from the values of n 1H , n 1H2 and t 1H2 for the 4-state system (Fig. 4(a)), and from the values of n U , n U U , and t U U for the molecular motor (Fig. 4(b)).For smaller errors, we can use a smaller constraint tolerance.
For both systems, smaller constraint tolerance leads to a better estimator as the value of the lower bound on the EPR approaches the true analytical value (Fig. 4(c) and (d)), demonstrating the importance of an accurate estimation of the observables.

D. Optimizing a simple model
Although our approach can be generalized to any number of hidden states, the analytical expressions for the observables become complicated, and the number of variables increases for a more complex coarse-grained topology.In turn, solving the optimization problem would require longer computation times.In order to test the performance of our estimator, we solved the optimization problem for a larger number of hidden states in a fully-connected network of 4, 5, and 6 states with only 2 observed states, assuming only 2 states are coarse-grained (Fig. 5(a)).Similarly, we tested the performance of our estimator for the case of the molecular motor with 2, 3, and 4 internal states at each physical position, assuming there are only 2. While generally, the estimator gives a more accurate result for the case of the 2 hidden state, which matches the assumption, it still provides a lower bound on the total EPR with comparable accuracy for a larger number of hidden states in the two systems (Fig. 5(b) and (c)).

IV. CONCLUSION
We present a new estimator for the entropy production rate, which gives a tight bound by formulating an optimization problem using both transitions and waiting times statistics.Our estimator can be applied to any system with known topology and it significantly surpasses previous estimators, as demonstrated for the two studied systems, the fully-connected hidden network, and the molecular motor.The variables for the optimization problem can be inferred from the observed statistics, where longer trajectories result in more accurate estimation and enable a smaller constraint tolerance value.Finally, for both systems, our approach can provide a lower bound on the total EPR for more complex systems, assuming a simpler underlying topology of the hidden states.Although we numerically showed that searching over all the systems with a simpler topology of the hidden part and the same observed statistics as the true system gave a lower bound on the total EPR for the two systems we studied, it remains an open problem to show this approach is universal.It would be interesting for future work to determine whether removing states from the hidden sub-network can only decrease the entropy production, given the observed statistics are conserved.
In summary, our approach is based on an optimization problem formulated using the observed statistics of a partially accessible system and provides a tight lower bound on the total EPR.The estimator can be used as a benchmark for comparing the performance of other estimators that rely on coarse-grained or partial information about the system.
The probability to observe a trajectory γ N : Since this is a convolution, we can perform a Laplace transform to get a simpler formula of multiplications of Laplace transforms of Markovian joint distributions of waiting times and transitions: In order to calculate the moments of the conditional waiting time distribution ψ IJK (t) for the coarse-grained state J conditioned on an initial state in I and a final state in K, our strategy is to calculate its Laplace transform ψIJK (s).We start by calculating ψiJk (s) which is the Laplace transform of the waiting distribution in coarse-grained state J, before jumping to a specific Markovian state k ∈ K, given it came from a specific Markovian state i ∈ I. Since we want the waiting time in J, we sum over all of the paths with any length N inside J with a final transition to k ∈ K, j 0 → j 1 → • • • → j N → k, weighed by the probability to jump from i ∈ I to the first state j 0 ∈ J: where ΨJJ (s) is a matrix of size N J × N J , and N J is the number of Markovian states inside J: As mentioned in the main text we denote ψiJK (s) ≡ k∈K ψiJk (s).Notice that ψiJK (s) is not normalized to 1 and it needs to be divided by ψiJK (s → 0), which is exactly the probability to jump from J to K, given the transition to J was from i.
This results from the fact that we used ψ ij (t), which is normalized to p ij .In order to get ψIJK (s), we sum ψNormalized iJK (s) over all of the Markovian states i ∈ I, weighed by the corresponding probability π i /π I of being in state i, given the system is in the coarse-grained state I: For a general probability density function f (t) : [0, ∞] → [0, 1] the Laplace transform is: and its k-th derivative by s is: Taking the limit s → 0: we find the k-th moment of the probability density function f (t): Therefore, the k-th moment t k IJK of the conditional waiting time distribution ψ IJK (t) is: Appendix C: Analytical expressions for the 4-state system The variables to consider for this system are the mass transition rates n ij and the steady-state probabilities π i for i, j ∈ {1, 2, 3, 4}, meaning a total of 16 variables.Note that π 1 , π 2 , n 12 and n 21 are fully observed.Therefore, we are left with 12 variables.With the following linear constraints, we can immediately reduce the problem to 6 variables.

Linear constraints
We impose probability conservation, mass transition rate conservation in the hidden Markovian states, and mass transition rate conservation between an observed Markovian state and the hidden coarse-grained state.

a. Probabilities
From conservation of the steady-state probability of the Markovian states within the coarse-grained hidden state:

Mass conservation at any Markovian state
We write the mass conservation for one of the hidden states (3 or 4), which for this system, is enough to guaranty the mass conservation for the other hidden state: c. First-order mass rates Here, we require the mass rate conservation of transitions in and out of the hidden state, providing 4 constraint equations:

Non-linear constraints
The second-order mass transition rates and the conditional waiting times moments can be expressed only as a non-linear function of the optimization problem variables.Here, we show the full derivations of these relations.
a. Second-order mass rates For this system, as mentioned in the text, we are interested in n 1H2 and n 2H1 , where the first and the last states are the observed Markovian states.From equation Eq.A1: Plugging into Eq.C4, we have: Remember we can express p ij in terms of the mass transition rates (Eq.3).

b. Conditional waiting time moments
We calculate the conditional waiting times moments t k iHj for i, j ∈ {1, 2}, in terms of the problem variables.Based on Eq.B14, we need to calculate ψNormalized iHj (s).
In order to get the expressions of the derivatives, we used the package Sympy in Python.

Appendix D: Analytical expressions for the molecular motor system
The variables to consider for the molecular motor system are the mass transition rates n u1 , n u2 , n d1 , n d2 , n l , n r and the steady-state probabilities π and π , meaning a total of 8 variables.With the following linear constraints, we can immediately reduce the problem to 4 variables.

Linear constraints
As in the 4-state system, we impose probability conservation, mass transition rate conservation in the Markovian states, and mass transition rate conservation for the observed transitions U and D.

a. Probabilities
From conservation of the steady-state probability of the Markovian states within the coarse-grained states:

Mass conservation at any Markovian state
We write the mass conservation for one of the hidden states (active or passive), which for this system, is enough to guaranty the mass conservation for the other hidden state: c. First-order mass rates Here, we require the mass rate conservation of transitions in and out of the coarse-grained state, providing 2 constraint equations: 2. Non-linear constraints Since we have 2 hidden states as in the 4-state system, the results from Appendix C can be used here.
a. Second-order mass rates We use the results for the 4-state system in Eq.C7, together with Eq.A2.For n U U , we need to sum over all the mass that goes up from the passive or active state, and then up again only to the passive state: For n DD , we need to sum over all the mass that goes down only from the passive state, and then down again to the passive or active state: We account for all of the transitions through a coarsegrained state i, and specify in the following calculations the Markovian state before jumping to i, and the following Markovian state, after state i, where i (i) denoted an active (passive) state.For example, (i − 1) − → (i + 1) represent two consecutive transitions, (i − 1) − → i − → (i + 1).Note that a transition upward is only to a passive state, so the previous state (being passive or active) in the first transition does not affect the waiting time.Furthermore, a transition downward is only from a passive state.
From Eq. B9: Now we calculate all the terms in the numerators, using Eq.C11 from the 4-state system results: ψ(i−1)→(i+1) (s) = 1 − w l w r (s + λ)(s + λ )  All of the denominators from Eq. D6 can be calculated by setting s → 0 in Eq.D7.Finally, we get the moments from equation Eq.B14.
In order to get the expressions of the derivatives, we used the package Sympy in Python.

FIG. 1 .
FIG. 1. Coarse graining.(a) The full Markovian system (left) and the coarse-grained system (right).(b) An example for a full trajectory (left) containing the actual states and the corresponding coarse-grained trajectory (right) containing only the observed states.

3 FIG. 3 .
FIG. 3. Molecular motor.(a) Illustration of the full molecular motor system including the coarse-graining of the active (red boxed square) and passive (ellipse) states.(b) Total EPR σtot (solid black line), our bound σ

FIG. 4 .
FIG. 4. Importance of data accuracy.(a) The error of some statistics of the 4-state system for different values of the trajectory length N .The absolute and relative errors are on the left and right axes, respectively.(b) The error of some statistics of the molecular motor system for different values of the trajectory length N .The absolute and relative errors are on the left and right axes, respectively.(c) The error of σ (1) opt results for the 4-state system for different constraint tolerance values, using the analytical statistics values.(d) The error of σ (1) opt results for the molecular motor system for different constraint tolerance values, using the analytical statistics values.Error bars stand for the standard deviation of 10 different realizations.

( 1 )FIG. 5 .
FIG. 5. Optimizing using a simple model.(a) Illustration of solving the optimization problem for a simple model with 2 hidden states (right), whereas the real system has more hidden states (left).(b) The results of σ (1) opt assuming the simple 4-state model (2 hidden states), when the real system has 2 (red cross), 3 (green triangle) or 4 (blue circle) hidden states.(c) The results of σ (1) n U U = n u1 (p u1 + p l p u2 ) 1 − p l p r + n u2 (p u1 + p l p u2 ) 1 − p l p r = (n u1 + n u2 )(p u1 + p l p u2 ) 1 − p l p r (D4)