Mechanisms and topology determination of complex chemical and biological network systems from first-passage theoretical approach

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1 Mechanisms and topology determination of complex chemical and biological network systems from first-passage theoretical approach Xin Li and Anatoly B. Kolomeisky Citation: J. Chem. Phys. 39, 4406 (203); doi: 0.063/ View online: View Table of Contents: Pblished by the AIP Pblishing LLC. Additional information on J. Chem. Phys. Jornal Homepage: Jornal Information: Top downloads: Information for Athors:

2 THE JOURNAL OF CHEMICAL PHYSICS 39, 4406 (203) Mechanisms and topology determination of complex chemical and biological network systems from first-passage theoretical approach Xin Li and Anatoly B. Kolomeisky,2,a) Department of Chemistry, Rice University, Hoston, Texas 77005, USA 2 Center for Theoretical Biological Physics, Rice University, Hoston, Texas 77005, USA (Received 30 Jly 203; accepted 23 September 203; pblished online 0 October 203) The majority of chemical and biological processes can be viewed as complex networks of states connected by dynamic transitions. It is fndamentally important to determine the strctre of these networks in order to flly nderstand the mechanisms of nderlying processes. A new theoretical method of obtaining topologies and dynamic properties of complex networks, which tilizes a first-passage analysis, is developed. Or approach is based on a hypothesis that fll temporal distribtions of events between two arbitrary states contain fll information on nmber of intermediate states, pathways, and transitions that lie between initial and final states. Several types of network systems are analyzed analytically and nmerically. It is fond that the approach is sccessfl in determining strctral and dynamic properties, providing a direct way of getting topology and mechanisms of general chemical network systems. The application of the method is illstrated on two examples of experimental stdies of motor protein systems. 203 AIP Pblishing LLC. [ I. INTRODUCTION Most chemical and biological systems can be described as complex networks of states. 5 To nderstand how these systems fnction one needs to determine the strctre and rates in these networks. In recent years, significant experimental and theoretical advances in investigating mechanisms of complex chemical and biological systems have been achieved. 6 5 It is now possible to monitor varios chemical and biological systems with a single-molecle precision and a high temporal and spatial resoltions. 6 Crrent theoretical methods also allow to calclate many moleclar properties with a high accracy. 2 5 However, a comprehensive determination of strctre-fnction relations for many systems remains an nsolved task. To nderstand the difficlty of this isse let s consider a general scheme of a chemical system that has an nderlying complex network of dynamically connected states see Fig.. A typical experimental measrement probes some complex event that starts from a state i and ends p at a state j (Fig. ). As a reslt, experiments prodce distribtions of these events, and mechanisms for nderlying chemical processes are proposed based on analysis of these data. However, this is an example of inverse problems that are difficlt to solve since more than one mechanism cold fit the measred distribtions. In addition, althogh all information abot mechanisms of complex processes is contained in these distribtions, in most cases only the first moments, mean dwell times, are analyzed. It has been recognized that stdies of distribtions of events for networks are associated with a first-passage proba) tolya@rice.ed lem in stochastic processes. 6, 7 The mathematical analysis of first-passage processes is well developed, 8, 9 however it is rarely sed for nderstanding complex systems. A more poplar approach is to tilize Monte Carlo compter simlations to generate distribtions and compare them with experiments. However, some minimal information on the strctre of the network is needed and a comptational cost becomes prohibitive for large networks. In many cases, more prodctive is a matrix method that has been sed in ion-channel measrements Bt it does not work for complex systems with mltiple transitions. Recently, a new absorbing bondary method has been proposed for nderstanding dynamics of moleclar motors. 23 Althogh the method is powerfl, it reqires some knowledge of the network topology and it is not efficient in obtaining average vales sch as mean times. 23 In this article, we develop a new method to predict strctral and dynamic information of the nderlying chemical and biological networks by analyzing distribtions of events withot assming any specific model. We propose the following hypothesis: dwell-time distribtions of events between two states as a fnction of time contain all information on the local strctre of the network and dynamic transitions between these states. The idea here is that at short times the distribtion is dominated by events that follow the shortest pathway (in terms of the nmber of intermediate states) between states. At transient times the effect of other pathways starts to bild p, and for large times dynamics is averaged over all possible pathways that reach the final state. Analyzing sch distribtions at all times for events between varios states shold provide a model-independent information on the whole network. In this work, the hypothesis is tested via exact analytical calclations and compter simlations for several types of networks. In addition, the method is applied for analyzing dynamics of two motor protein systems /203/39(4)/4406/9/$ , AIP Pblishing LLC

3 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203) FIG.. A schematic view of a general chemical network. Nodes correspond to individal states and arrows describe possible transitions. Experimental measrements typically provide the information on distribtion events that start at some initial state i and finish at some final state j. II. RESULTS A. First-passage analysis of dwell-time distribtions for homogeneos One-Dimensional (D) networks We consider first a simplified D network of discrete states j = 0,..., N, as shown in Fig. 2(a). For real processes it corresponds to dynamics of motor proteins, 2, 24 polymer translocation, 7 or protein folding. 6 Sppose that at time t = 0 the molecle is in a state j = m. The system can transfer to the FIG. 2. A schematic view of network systems: (a) a seqential D network; (b) D network with irreversible detachments; (c) D network with a branch; (d) a network with two parallel segments. state j = m + witharate m, while the backward transition rate to the state j = m is given by w m. We are interested in obtaining a distribtion fnction that describes reaching the left end state (j = 0) or the right end state (j = N) for the first time. Specifically, a first-passage probability fnction f N, m (t) is defined as a probability that the system starting at time t = 0 in the state m will reach the right end state N at time t for the first time before reaching the left end state j = 0. In a similar way we define a first-passage probability fnction f 0, m (t) to reach the left end. Let s concentrate on the f N, m (t) fnction, and all reslts for f 0, m (t) can be obtained from symmetry argments (changing m N m and m w m ). The temporal evoltion of these probability fnctions can be described by backward master 6, 7 eqations, f N,m (t) t = m f N,m+ (t) + w m f N, (t) with the following bondary conditions: ( m + w m )f N,m (t), () f N,0 (t) = 0, for all t, f N,N (t) = δ(t), (2) f N,m (0) = 0, for m N. The main idea of or approach is to tilize Laplace transforms of distribtion fnctions defined as f N,m (s) = De to the bondary conditions we have 0 f N,N (s) =, e st f N,m (t)dt. (3) f N,0 (s) = 0. (4) To illstrate the power of the method consider a simpler sitation with homogeneos transition rates, m = and w m = w for all m. Then the backward master eqation () in the Laplace presentation can be written as s N,m = N,m+ + w N, ( + w) N,m. (5) One cold define an axiliary fnction a(s) = s + + w, and Eq. (5) simplifies into the following: a N,m = N,m+ + w N,. (6) This eqation can be easily solved by assming that the general form of the soltion is N,m = Ax m with nknown parameters A and x to be fond from the sbstittion of this ansatz into Eq. (6). It leads to a very compact formla, where x = a + a 2 4w 2 f N,m (s) = xm xm 2 x N xn 2, (7), x 2 = a a 2 4w. (8) 2 For a special case of w = 0 one can show that x = s+ and x 2 = 0, yielding ( ) N m N,m (s) =, (9) s +

4 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203) which can be easily inverted to obtain the expected firstpassage distribtion fnction f N,m (t) = N m t N e t. (0) (N m )! The advantage of sing Laplace transforms is that all dynamic properties at all times can be calclated explicitly. For example, the overall probability to reach the state N, known as a splitting probability, 6, 7 is given by N,m = 0 f N,m (t)dt = f N,m (s = 0). () The conditional mean first-passage time, which is measred in experiments as a mean dwell time, is defined as τ N,m = τ m(), with τ m () = d N,m (s) s=0. (2) N,m ds Similar expressions can be obtained for any other dynamic properties of the system. For simple D networks with homogeneos transition rates in one direction m = and no backward transition w m = 0, a simple expression for the first-passage distribtion fnction at early times can be obtained from Eq. (0), ln f N,m (t) (N m ) ln t + C, (3) where the exponential part in Eq. (0) is close to and can be neglected at t and C is a time-independent constant. This is an important reslt since it shows a linear relation (on loglog scale) between the first passage distribtion and time, and the slope gives the nmber of intermediate states between the initial and final states. It sggests that dwell-time distribtions can be sed as a tool for determining the network strctre. To prove this idea frther, compter simlations have been performed to model the homogeneos D network with transition rates m = = 5.0 s and w m = 0. The firstpassage distribtions are shown in Fig. 3(a) for all times and in Fig. 3(b) at early times for different initial states m 8 with final state N = 0. For the special case with m = 9, the distribtion gives an exponential decay as expected for the absence of intermediate states. As shown in Fig. 3(a) the analytical predictions from Eq. (0) [dashed lines] agree perfectly with the simlation reslts. The solid lines in Fig. 3(b) are given by linear fittings of simlation data at small times and the slope of the line (on log-log scale) goes down as the nmber of intermediate states between the initial and final states decreases. The slopes are very close to expected vales (N m ), and ronding to the pper integer for each slope gives the exact nmber of intermediate states. The reason for a slightly smaller vale of the slope for each crve from the fittings is the effect of the exponential term that contribtes more to the distribtion with increasing time. Expressions similar to Eq. (0) have been proposed earlier for simple networks as in Fig. 2(a), 25 where a pertrbation theory was applied with transitions from the probed states ignored and only an approximate expression was obtained. Here, in or work an exact expression for the first-passage distribtion fnctions is derived withot any assmptions, and more general and realistic networks are discssed. The early temporal behavior of the distribtion fnction can also be obtained from the limiting s behavior of corresponding Laplace transform. Then, for the homogeneos case from Eq. (7) with m = 0, and w m = w 0, it can be shown that f N,m (s) which corresponds to ( s + + w ) N m, (4) f N,m (t) N m t N e (+w)t (5) (N m )! at early times. Again, the strctral information of the network for simple D networks with homogeneos transition rate is contained in the power law dependence of the firstpassage distribtion which can be obtained from a simple analysis of distribtion fnction at early times. B. First-passage analysis of dwell-time distribtions for inhomogeneos one-dimensional networks Or method can be sed for arbitrary networks, and it will be shown analytically by considering a general inhomogeneos D network system in Fig. 2(a). In this case, the backward master eqations in the Laplace presentation are given by a m N,m = m N,m+ + w m N,, (6) (a) (b) a f N, = f N,2, (7) FIG. 3. First passage distribtions for a homogeneos D network with only forward transitions. The position of the initial state m is varied from to 8 and the final state is N = 0. (a) First-passage distribtions at all times. (b) First-passage distribtions at early times. Symbols correspond to compter simlations. The dashed lines in (a) are given by the analytical expression from Eq. (0), while the solid lines in (b) are from linear fittings of the simlation data. The slope for each crve in (b) is also indicated. a N N,N = N + w N N,N 2, (8) where a m = s + m + w m. (9) There are no known exact soltions for inhomogeneos D networks, bt stimlated by reslts for homogeneos systems, we propose the following ansatz for Laplace transforms of the distribtion fnctions: K=0 N,m (s) = α K(m)s K N K=0 α K(N)s, (20) K

5 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203) where α K (m) are nknown coefficients. It sggests that the Laplace transforms of distribtion fnctions can be viewed as ratios of two polynomials with respect to the variable s. Several examples to spport this are provided in the Appendix. It is important to note that the coefficients α K (m) can be determined self-consistently from Eq. () which allows s to determine explicitly all dynamic properties of the system. Frthermore, the limiting behavior of distribtion fnctions (for s and s 0) can be obtained knowing only few of coefficients α K (m), which significantly simplifies calclations. For example, one can show (see the Appendix) that splitting probabilities and mean dwell times to reach the final state N are given by N,m = α 0(m) α 0 (N),τ N,m = α (N) α 0 (N) α (m) α 0 (m). (2) For D networks with inhomogeneos rates [see Fig. 2(a)] the analysis yields a power law dependence of firstpassage distribtions at early times (see the Appendix for details), N j=m f N,m (t) (N m )! t N. (22) This reslt flly agrees with the hypothesis, allowing to determine the nmber of intermediate states. Note also, that the prefactor in Eq. (22) contains the information on forward transition rates, and the exponent does not depend on transitions rates. The relative vales of transitions rates only specify the range of the applicability of Eq. (22). To test the predictions from Eq. (22) compter simlations have been performed. The first-passage distribtions are presented in Fig. 4(a) for all times and in Fig. 4(b) at early times for different initial states m and the final state N = 0. The transition rates w m (a) (c) FIG. 4. First passage distribtions for inhomogeneos D networks. The initial state m is varied from to 8, and the final state is N = 0. Set of random rates is sed for simlations. (a) and (b) Distribtions for D inhomogeneos network shown in Fig. 2(a). (c) and (d) Distribtions for D network with irreversible detachments as shown in Fig. 2(b). (a) and (c) First-passage distribtions for all times. (b) and (d) First-passage distribtions at early times. Symbols correspond to compter simlations. The solid lines in (b) and (d) are from linear fittings of the simlation data. The slope for each crve in (b) and (d) is also indicated. (b) (d) and m are randomly chosen. The solid lines in Fig. 4(b) come from linear fittings of the simlations at small t. Again, a decrease of the slope of the distribtion fnction with smaller nmber of intermediate states is observed, as predicted by Eq. (22). Ronding the fitted vales of the slope to the pper integer gives the nmber of intermediate states. The analysis can also be extended to another limit when t, with the first-passage distribtion prodcing (see the Appendix) f N,m (t) N,m exp( t/τ N,m ). (23) τ N,m It shows how dynamic properties of networks can be obtained from large-time behavior of dwell-time distribtions. It is important to note that these properties are averaged over many pathways connecting initial and final states so that the important information on the networks can also be obtained in this limit. C. Effect of irreversible transitions To show the applicability of or method for more realistic systems consider a D network with irreversible detachments as illstrated in Fig. 2(b). This sitation corresponds to florescence measrements of protein folding 6 when photon blinking phenomena can be viewed as irreversible dissociations from the network. Defining δ m as an irreversible detachment rate from the state m, we can write backward master eqations for evoltion of first-passage probabilities f N, m (t)in the following form: f N,m (t) = m f N,m+ (t) + w m f N, (t) t ( m + w m + δ m )f N,m (t), (24) with corresponding bondary conditions given by Eq. (2), as explained above. We start first with a simple case of D networks with constant irreversible detachments, i.e., all detachment rates are eqal δ m = δ. The main difference of the networks with irreversible detachments in comparison with simple D networks withot detachments is the fact the probabilities f N, m (t) are not conserved. This sggests the way to obtain analytic soltions by mapping the model with detachments into the model withot detachments. It can be done by defining a first-passage probability fnction for srviving particle that is a conserved qantity, F N,m (t) = f N,m (t)e δt. (25) Sbstitting this into the backward master eqations (24) we obtain the eqation for F N, m (t), F N,m (t) = m F N,m+ (t) + w m F N, (t) t ( m + w m )F N,m (t), (26) which is identical as Eq. () describing an inhomogeneos system withot detachments. The Laplace transforms of the original and srviving first-passage probabilities are related as f N,m (s) = F N,m (s + δ). (27)

6 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203) Using this approach the model can be solved analytically and all dynamic properties can be calclated. The asymptotic behavior of first-passage distribtion fnction for t 0isgiven by N f N,m (t) = F N,m (t)e δt j=m (N m )! t N e δt N j=m (N m )! t N, (28) which is same as Eq. (22). Therefore, the early temporal behavior of the first-passage distribtions is not modified by the irreversible dissociation processes with a constant detachment rate. For a more general case with position dependent dissociation rates it has been shown earlier by one of s that the model with the irreversible detachments can be mapped into the model withot detachments by tilizing a matrix renormalization approach. 29 Calclations indicate that the power law dependence of first-passage distribtion fnction similar to Eq. (22) is again fond at early times (see the Appendix for details). This power law dependence is also spported by compter simlations as indicated in Figs. 4(c) and 4(d). These reslts for D networks with detachments clearly spport or hypothesis that the nmber of intermediate states in complex networks can be determined by analyzing dwell-time distribtions at t 0. D. Analysis of first-passage events for networks with different topology In order to demonstrate the application of the method to more complex systems, two network models with different topology, as presented in Figs. 2(c) and 2(d), are also analyzed. One of the systems (Fig. 2(c)) has a set of states that branches ot of the main pathway, while another system (Fig. 2(d)) has two parallel segments of states. For these two cases analytical reslts cannot be obtained easily, so we mainly focs on compter simlations. The first-passage distribtion fnction for the D network with the branched states are shown in Figs. 5(a) and 5(b) for different initial states. The branch with two states is connected to the state m = 5on the main pathway. One cold see that for m 5 there are several distinct pathways. However, distribtion fnctions again have a power law dependence at early times (see Fig. 5(b)), and the slopes are given by the nmber of intermediate states in the shortest pathway to the final destination. The presence of the branched states does not affect the network dynamics at early times, while the effect of the branch can be seen on long-time behavior of distribtion fnctions (compare Figs. 4(a) and 5(a)). For the network system with two parallel segments (Fig. 2(d)) we assmed that the loop starts at the site m = 3, and then two different cases of having 2 or 3 states in the pper segment have been analyzed. First-passage distribtion fnctions for initial states m = 2, 3, and 4 and the final state N = 0 have been compted and presented in Figs. 5(c) and 5(d). As for all considered networks, the power law dependence of the distribtion fnctions at early times (a) (c) FIG. 5. First passage distribtions for network models with complex topology. (a) and (b) First-passage distribtions for a network with one branch as shown in Fig. 2(c). (c) and (d) First-passage distribtions for a network with two parallel segments as shown in Fig. 2(d). (a) and (c) First-passage distribtions in fll time regime. (b) and (d) First-passage distribtions at early times. The distribtion fnctions for different initial states with m = 5 and the final state N = 0 are shown. For the network with two parallel segments the distribtion fnctions for initial states with m = 2 4 and the final state N = 0 are shown. Symbols correspond to compter simlations. Set of random rates is sed for simlations. The solid lines in (b) and (d) are from linear fittings of the simlation data. The slope for each crve in (b) and (d) is also indicated. is observed (see Fig. 5(d)), and the slopes yield the nmber of intermediate states on the shortest pathway between initial and final states. The loop topology does not modify network dynamics at short times, while the effect of the network strctre can only be seen at large times since in this case the dwell-time distribtions are obtained by averaging over of all possible pathways between initial and final states. These reslts for both networks with complex geometry again spport the hypothesis. E. Application of the method for stdying motor protein dynamics It is important to test or method by analyzing real processes, and two motor protein systems 26, 27 have been chosen as examples. Myong et al. 26 have investigated the DNA nwinding by virs NS3 helicase sing single-molecle florescence analysis. Discrete steps of 3 base pairs (bp) have been observed and the corresponding dwell time histograms were also obtained with a rising phase at the beginning followed by a phase with decay. These observations indicate that a 3-bp transition is not the elementary step dring the DNA nwinding. The distribtions at the rising phase are shown in Fig. 6 (as demonstrated by circles and triangles). They can be fitted by linear crves (on log-log scale) as indicated in Fig. 6, spporting the power law dependence at early times. Or method predicts that the slopes of these crves give the nmber of intermediate states in the shortest pathway. The obtained vales,.89 and.49, indicate the existence of two more hidden biochemical states dring the nwinding of DNA. Then one concldes that the 3-bp transition is very likely composed of (b) (d)

7 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203) FIG. 6. Dwell time distribtions for myosin-v s 74-nm steps (sqares) from Ref. 27 and for steps in DNA nwinding by helicase NS3 (circles and triangles for different lengths of DNA) from Ref. 26 at early times. The lines are linear fits of experimental observations on log-log scale. The slopes for each crve are also indicated in the figre. three elementary steps, and it is tempting to assign each step with a bp length. However, one has to be carefl here since this analysis only provides the nmber of intermediate states and not their spatial positions. Myong et al. 26 has proposed a similar mechanism of DNA nwinding by helicase with bp per step bt their analysis is based on a less realistic model that assmes irreversible eqal-rate transitions. In contrast, or method is model independent. A second example is related to stdies of myosin-v molecles. Myosin-V are dimeric motor proteins that walk along actin filaments with a step size of 37 nm. 28 Two alternative stepping models have been proposed. 27 In the inchworm mechanism the advancement of one motor head is followed by the motion of the second one. In the hand-overhand mechanism each motor head makes a 74 nm step in the alternating fashion. The dynamics of myosin-v has been stdied by attaching a single florescent label to different parts of the protein molecle. 27 For a dye close to one motor head, the two different stepping mechanisms either give niform 37 nm steps for the inchworm model or alternating 74 nm and 0 nm steps for the hand-over-hand model. Experiments have fond only 74 nm steps and a dwell time distribtion for these steps was also obtained, prodcing a rising phase at the beginning followed by a phase with decay. The distribtions measred in Ref. 27 at the rising phase is shown as open sqares in Fig. 6. Again, the distribtions were fitted by a linear crve (on log-log scale) with a slope of 0.82, as shown by the solid line in Fig. 6. The slope vale is close to, indicating the existence of one more intermediate state corresponding to the motion of the other motor protein head that is not florescently labeled nder the hand-over-hand mechanisms. III. DISCUSSION A new theoretical approach to predict the strctral and dynamic properties of complex chemical and biological networks via first-passage analysis of data on distribtions of events is developed. It is based on the hypothesis that the first-passage distribtions of events are flly specified by local topology and dynamics of nderlying networks. Exact analytical calclations and Monte Carlo simlations are performed for several networks. A power law dependence of first-passage distribtion at early times is fond in all cases, and the corresponding exponent reflects the nmber of intermediate states in the shortest pathway between the initial and final states. At large times or approach predicts that distribtions decay exponentially as a fnction of time with parameters that depend on the probability and the mean dwell time to reach the final state. The microscopic origin of the method can be explained sing the following argments. In the network at early times only the fastest events are recorded and they proceed via the shortest link between initial and final states. Ths the first-passage distribtion at early times corresponds to trajectories along the shortest path in terms of the nmber of intermediate states. In contrast, the dynamics of reaching the final state at large times is averaged ot over many pathways connecting two states. Using this approach, or analysis of short-time arrival dynamics from distribtion fnctions indicates the existence of two intermediate biochemical states for each DNA nwinding step by NS3 helicase and one intermediate state for stepping of myosin- V molecles, spporting the hand-over-hand stepping mechanisms. In principle, or approach is capable to determine fll strctral and dynamic information of networks if experimental information on distribtions of events is available. To obtain the fll strctre many events with varying initial and final states mst be analyzed. However, in the application of the method there are still qestions. In this work mostly the information on short-time dynamics of distribtion fnctions has been tilized. We arged that the behavior of distribtion fnctions at large times provides the information on mean dwell times and probabilities to reach the final state. However it is not clear how to obtain explicit information from these average properties. In addition, more theoretical stdies are needed to clarify how distribtion of events depends on the nderlying topology of the network. Despite these isses, it seems that the presented method might be a powerfl tool for ncovering mechanisms of complex processes. Recently, a new method to investigate similar problems, which is based on information theory approach, was proposed. 30 However, this method was only employed for very simple networks described by mlti-exponential dwell time distribtions and only in the limit of large times. Frthermore, only approximate vales for the nmber of hidden states cold be obtained. In contrast, or method is mch simpler and it provides an exact nmber of hidden states from early times. ACKNOWLEDGMENTS This work was spported by grants from National Instittes of Health (Grant No. R0GM ) and the Welch Fondation (Grant No. C-559). APPENDIX: DETAILED THEORETICAL CALCULATIONS This appendix consists of several parts with detailed derivations of the eqations and relations sed in the main text.

8 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203). Inhomogeneos D networks The backward master eqations for a general inhomogeneos D network model presented in Fig. 2(a) in the Laplace presentation are given by Eqs. (6) (8) in the main text. In order to solve these eqations we sggest the following ansatz for Laplace transform of distribtion fnction as explained in the main text: K=0 N,m (s) = α K(m)s K N K=0 α K(N)s, (A) K where α K (m) are some nknown coefficients. Eqation (A) satisfies bondary conditions for m = N. It sggests that the Laplace transforms of distribtion fnctions can be viewed as ratios of two polynomials with respect to the variable s. For example, for the simplest case N = 2 and m = it can be shown that 2, (s) =, (A2) s + + w which agrees with the ansatz [see Eq. (A)]. Inverting this expression yields the corresponding first-passage distribtion fnction, f 2, (t) = exp[ ( + w )t]. (A3) Also for the case N = 3 with m = orm = 2 one can find 2 3, (s) =, s 2 + s( w + w 2 ) w 2 + w w 2 (A4) and 2 s w 2 3,2 (s) =, s 2 + s( w + w 2 ) w 2 + w w 2 (A5) which also agree with the ansatz [see Eq. (A)]. These expressions can also be inverted analytically. The coefficients α k (m) are determined self-consistently from backward master eqations, which provides a direct way to obtain explicitly all dynamic properties of the system. For a general case with m =, the expression for Eq. (A) is given by α 0 () N, (s) = N K=0 α K(N)s. (A6) K Then, combining this relation with Eq. (7) in the main text we obtain α 0 ()(s + + w ) = (α 0 (2) + α (2)s). (A7) It can be proved for general conditions that α 0 () =, which simplifies many calclations. Specifically, from Eq. (A7) we derive α (2) = /, α 0 (2) = + w. (A8) Similarly, for a general m, one can obtain the following expressions: α 0 (m) = + K w j K= j= j, (A9) α (m) = j=. (A0) Using Eqs. (6) and (A) and considering only terms proportional to s leads to α 0 (m) + α (m)( m + w m ) = m α (m + ) + w m α (m ). (A) Let s define a new axiliary fnction, then it can be shown that () m = α (m + ) α (m), (A2) α 0 (m) = m () m w m (), (A3) where α 0 (m) are known from Eq. (A9). The general soltion of Eq. (A3) is given by m () m = α 0(K) m w j, (A4) K l= K= l= j=k+ which leads to l α (m) = () l = α 0(K) K K= l w j j=k+ j. (A5) Similarly, from Eqs. (6) and (A), considering the coefficients for the terms proportional to s K, one finds α K (m) + α K (m)( m + w m ) = m α K (m + ) + w m α K (m ). (A6) Using the same procedre as above, the expression for the coefficient α K (m) can be written as i α K (m) = α K (l) i w j. (A7) l i=k l=k j=l+ This reslt is important since it indicates that the coefficient α K (m) can be determined from α K (m). Therefore, all coefficients α K (m) can be calclated iteratively becase we already determined the expressions for α 0 (m). Then the Laplace transform N,m (s) of distribtion fnction can be derived from Eq. (A) and the distribtion fnction can be obtained from the inverse Laplace transforms. In fact, to calclate many dynamic properties, we do not need to know all coefficients α K (m). For example, the splitting probability N, m is simply given by N,m = N,m (s = 0) = α 0(m) α 0 (N) = + + N K w j K= j= K= K j= w j, (A8) which agrees with known expressions in the literatre. 6 For D networks with homogeneos transition rates w m = w and

9 X. Li and A. B. Kolomeisky J. Chem. Phys. 39, 4406 (203) m = m, the splitting probability can be written in the simpler form, N,m = N,m (s = 0) = γ m γ, (A9) N where γ = w. From Eq. (2) the general expression for the mean firstpassage time to exit at the site N is given by τ m () = tf N,m (t)dt = d N,m (s) s=0 ds 0 = α 0(m)α (N) α (m)α 0 (N). (A20) α 0 (N) 2 And for the conditional mean first-passage time we obtain τ N,m = τ m() = α (N) N,m α 0 (N) α (m) α 0 (m). (A2) For the special case with w j = 0 from Eq. (A9) one can show that all coefficients α 0 (m) aregivenby α 0 (m) = α 0 () =. (A22) From Eq. (A8) the splitting probability is given by N, m =. The coefficient α (m) can be then written as α (m) = i= i, which gives the conditional mean first-passage time τ N, m, τ N,m = α (N) α (m) = N i=m i. (A23) (A24) Let s consider another special case with w = 0 that corresponds to a reflecting bondary condition. It can be shown that in this case the coefficient α 0 (m) is eqal to α 0 (m) =, for all m. And the coefficient α (m)isgivenby with () K α (m) = () K, K = i i= K= K w j j=i+ j. (A25) (A26) (A27) Therefore, the conditional mean first-passage time τ N, m can be written as N K τ N,m = K w j. (A28) K=m i= i j=i+ j From simple D networks with inhomogeneos transition rates, it is sefl to consider the asymptotics for f N,m (s). For s, which corresponds to t, we have N,m (s) α (m) α N (N) sm N = = N j=m s N m, j= N j= s N m (A29) where Eq. (A0) was sed to obtain the final expression above. Then, the first-passage distribtion fnction at small t 0 is given by N j=m f N,m (t) (N m )! t N, (A30) which is Eq. (22) in the main text. Again, we obtained a power law dependence of the first-passage distribtion at early times. For another limit with s 0ort, we find N,m (s) α 0(m) + α (m)s α 0 (N) + α (N)s α 0(m) α 0 (N) [ + α (N) α 0 (N) α (m) α 0 (m) ], s (A3) which leads to the first-passage distribtion fnction for t as f N,m (t) N,m e t τ N,m, (A32) τ N,m where N, m and τ N, m are given by Eqs. (A8) and (A2), respectively. The eqation above corresponds to Eq. (23) in themaintext. 2. Effect of irreversible transitions For D networks with irreversible detachments from the main pathway, as illstrated in Fig. 2(b), the general backward master eqations for evoltion of first-passage probabilities of reaching the state N for the first time at time t starting from the state m at t = 0aregivenbyEq.(24) in the main text. For a general inhomogeneos case of D networks with dissociations, it has been shown earlier that the model with irreversible detachments can be mapped into the model withot detachments by tilizing a matrix renormalization approach. 29 We can define a first-passage probability fnction F N, m (t)as F N,m (t) = f N,m (t)e λt+γ m, (A33) where λ and γ m will be fixed so that F N, m (t) satisfies the renormalized, backward master eqations F N,m (t) = ũ m F N,m+ (t) + w m F N, (t) t (ũ m + w m )F N,m (t), (A34) with properly renormalized rates ũ m and w m. By sbstitting Eq. (A33) in Eq. (24) one can obtain F N,m (t) = m e γ m γ m+ F N,m+ (t) t + w m e γ m γ F N, (t) ( m + w m + δ m λ)f N,m (t). (A35)

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