A Unified Approach to Gated Reactions on Networks
AA Unified Approach to Gated Reactions on Networks
Yuval Scher , ∗ and Shlomi Reuveni , † School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry,and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel (Dated: January 14, 2021)For two molecules to react they first have to meet. Yet, reaction times are rarely on par with the first-passagetimes that govern such molecular encounters. A prime reason for this discrepancy is stochastic transitions be-tween reactive and non-reactive molecular states, which results in effective gating of product formation andaltered reaction kinetics. To better understand this phenomenon we develop a unifying approach to gated re-actions on networks. We first show that the mean and distribution of the gated reaction time can always beexpressed in terms of ungated first-passage and return times. This relation between gated and ungated kineticsis then explored to reveal universal features of gated reactions. The latter are exemplified using a diverse set ofcase studies which are also used to expose the exotic kinetics that arises due to molecular gating.
The first-passage properties of a random walker are centralto the study and analysis of stochastic phenomena [1–4]. Theclassic problem of a random walker in search of a target arisesnaturally in a variety of fields, be it biology [5–13], finance[14, 15] or network science [16–21], to name a few. Markedly,the first-passage problem is central to the theory of diffusionlimited reactions, where a reaction between two species canbe modeled as stochastic motion that terminates on contact[22–24]. However, while having two molecules at the sameplace and at the same time is a necessary condition for theoccurrence of a chemical reaction, more is oftentimes requiredfor the reaction to actually take place.It has long been realized that in order to better depict chem-ical reactions one has to consider the possibility of infer-tile molecular collisions [25–35]. This realization later ma-tured to the concept of gated reactions, which occur onlywhen molecules collide while in a reactive state (alternatively,the gate is open) [36–51], see Fig. 1. A similar idea of agated boundary was applied in narrow gated-escape problems[52, 53] and search for gated targets [54, 55]. Depending onthe context, the first-time of arriving at the boundary whileat the reactive state is interchangeably referred to as reactiontime, first-absorption time or first-hitting time. Throughoutthe years many examples of gated processes were explicitlysolved for, but a general understanding of gated-reaction ki-netics is still missing. A notable step in that direction is theformalism developed by Spouge, Szabo and Weiss [44].At the heart of this letter is a renewal approach that is usedto build a unified framework to gated reactions on networks.Specifically, by employing this renewal approach we providefor simple and general relations between gated and ungated re-action times. These, in turn, are used to show that it is enoughto solve for the ungated problem to readily obtain a solutionfor the corresponding gated problem. Solving the ungatedproblem is generally a much simpler task, which e.g., allowsfor clean re-derivation of classic results that were previouslyobtained via brute-force methods. Moreover, the relations de-rived below open the door for systematic and widespread anal-ysis of gated kinetics by providing ready-made solutions inall cases where the underlying, i.e., ungated, problem has al-ready been (or can be) solved. This important feature of our
FIG. 1. Illustration of a gated reaction on a network. The reac-tant, here depicted as a small sphere, performs a random walk whileswitching between the non-reactive (NR, blue) and reactive (R, red)states. Reaction occurs when two conditions are met: (i) the reac-tant is at the target (shown as bullseye); and (ii) the reactant is in thereactive state. framework has already proven extremely useful in the contextof stochastic resetting where similar renewal methods wereheavily employed [56–72].The framework developed herein is also used to unravelthe existence of universal features of gated reactions. Specifi-cally, we point to the effect of fluctuations in the time betweenconsecutive molecular collisions and show that these alwayslead to an increase in the mean completion time of a gatedreaction. Moreover, in cases where wild fluctuations lead todiverging means, we show that the exponent governing theheavy-tailed asymptotics of the ungated reaction is universallyinduced upon the gated reaction; thus generalizing earlier ob-servations made for simple diffusion [41, 42, 55]. Finally,the framework is utilized to expose exotic kinetic features thatarise due to gating. As we show, these can be observed evenin simple model systems such as the 1d random walk, yet theyhave so far been overlooked. Recent advancements in single-molecule technologies make us hopeful that predictions com-ing from our framework will soon be tested experimentally. a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Gated reactions on networks. —The set of problems weanalyze below is formulated as follows: Consider a reac-tant (particle) performing a continuous-time random walk(CTRW) [73] on a network (see Fig. 1, for an example ofa finite network embedded in a 2D lattice). In addition to itsspatial motion, the reactant also undergoes stochastic internaldynamics flipping between a reactive state and non-reactivestate. A reaction is deemed to occur if the reactant is foundin the reactive state while its spatial position overlaps with theorigin — a designated target site which, e.g., harbours a com-plementary reactant. To keep things general, we make no as-sumptions on the network on top of which the dynamics takesplace, the distributions that govern waiting times in the differ-ent sites, and the distributions that govern the random motion(jumps) of the particle between network sites.To progress, we assume that the position of the particle isdecoupled from its internal state. The strategy is then to “di-vide and rule”. First, we tackle the reactant’s internal dynam-ics which is hereby assumed to be governed by a continuous-time Markov chain composed of two states: Reactive (R)and Non-Reactive (NR). The transition rate from R to NRis denoted by α and the transition rate from NR to R is de-noted by β (Fig. 1, top-left). We are interested in the condi-tional probability to be in each of the states after time t , givenan initial internal state q ∈ { R, NR } . For example, when q = NR, this is given by P ( R , t | NR ) = π R ( − e − λ t ) andP ( NR , t | NR ) = π NR + π R e − λ t , where λ = α + β is the ef-fective relaxation rate and with π R = β / λ and π NR = α / λ standing for the equilibrium occupancies.Consider now the time T ( x ) it takes the particle to re-act given an initial location r and an initial internal state q which we jointly denote by x = ( r , q ) . Two contributionsfeed into T ( x ) as illustrated in Fig. 2: (i) T FP ( r ) whichis the first-passage time of the particle to the origin; and (ii)the time it takes the particle to react after it has first reachedthe origin. If the particle arrives at the origin in the reactivestate it reacts immediately. Otherwise, the particle—which isnow found at the origin in the non-reactive state—can be seento start its motion anew with the following initial conditions NR ≡ ( , NR). Thus, letting T ( NR ) denote the reaction timestarting from NR , we have T ( x ) = T FP ( r ) + I FP T ( NR ) , (1)where I FP is an indicator random variable that receives thevalue 1 if the particle first arrived at the origin in the non-reactive state and 0 otherwise.Taking expectations in Eq. (1) we find [74] h T ( x ) i = h T FP ( r ) i + h π NR ± ( − π q ) ˜ T FP ( r , λ ) i h T ( NR ) i , (2)where we have a plus sign if q = NR, and a minus sign if q = R, and where ˜ T FP ( r , λ ) = h e − λ T FP ( r ) i is the Laplacetransform of T FP ( r ) evaluated at λ . Note that in the limit λ h T FP ( r ) i (cid:29)
1, i.e., when r is far enough such that the in-ternal state equilibrates before the particle arrives at the origin, FIG. 2. The gated reaction time T ( x ) from Eq. (1) has two con-tributing factors: (i) the first-passage time of the particle to the origin T FP ( r ) ; and (ii) the time it takes the particle to react after it has firstreached the origin. In the illustration this time is given by T ( NR ) ,and I FP in Eq. (1) takes care of situations where the particle arrives atthe origin in the reactive state and reacts immediately. On the time-axis, J stands for jump and T for transition between internal states. the Laplace transform is negligible and the mean reaction timein Eq. (2) becomes independent of the initial internal state.The distribution of the gated reaction time can also be com-puted. Taking the Laplace transform of Eq. (1), we find [74]˜ T ( x , s ) = π R ˜ T FP ( r , s ) + π NR ˜ T FP ( r , s ) ˜ T ( NR , s ) ± ( − π q ) ˜ T FP ( r , s + λ ) h ˜ T ( NR , s ) − i , (3)where we have a plus sign if q = NR, and a minus sign if q = R. Note that here too, the above expression simplifiesconsiderably in the limit λ h T FP ( r ) i (cid:29) T FP ( r , s + λ ) is negligible, and one is left with the first two terms whichare independent of the initial internal state.Examining Eqs. (2) and (3) above reveals that part of thework required to determine the mean and distribution of agated reaction time can be reduced to the solution of a stan-dard (ungated) first passage time problem. Namely, one re-quires T FP ( r ) which can be obtained by standard methods[4]. However, we are still left with the task of computing thegated reaction time T ( NR ) . The mean and distribution of thisrandom time will be our main focus going forward.Let W denote the random time the particle waits at the ori-gin before jumping to a different site, and let X stand for therandom location of the particle following this jump. Setting W NR as the random waiting time of the particle in the non-reactive state, we observe that the gated reaction time T ( NR ) can be written as follows T ( NR ) = (cid:26) W NR if W NR < W , W + T FP ( X ) + I FP T ( NR ) if W NR ≥ W , (4)where T FP ( X ) is the first-passage time to the origin startingfrom X , I FP is defined as before, and T ( NR ) is an IID copyof T ( NR ) . Indeed, if W NR < W , the particle transitions tothe reactive state before jumping out of the origin and a reac-tion immediately follows. Conversely, if W NR ≥ W , the parti-cle jumps out of the origin before transitioning to the reactivestate. It then requires a time T FP ( X ) to return to the origin,and an additional gated reaction time T ( NR ) provided it re-turned in the non-reactive state. We thus see that the gatedreaction time T ( NR ) can be expressed in terms of an ungatedfirst-passage time T FP ( X ) and the waiting times W NR and W . FIG. 3. Fluctuations in the return time to the origin increase the mean completion time of a gated reaction. Panel a). The mean reactiontime h T ( NR ) i from Eq. (5) vs. the internal relaxation rate λ = α + β . Here we consider the symmetric case α = β = λ / γ = T FP ( X ) from a Gamma distribution with unit mean and increasing values of the coefficientof variation CV (see [74] for details). The black line corresponds to the deterministic, CV =
0, case which gives rise to the lower bound inEq. (6). Higher CV s lead to higher mean completion times as predicted. It can be appreciated that the asymptotic h T ( NR ) i ∼ β − ∼ λ − behaviour is common to all curves in both the high and low λ regimes [see Eq. (5) and discussion below]. Panel b). The mean reactiontime normalized by its lower bound: emphasizing that the behaviour at intermediate λ s is sensitive to the CV of the return time to the origin.Furthermore, each curve has a unique value λ ∗ for which the deviation from the lower bound is maximal. This value can be determined bysolving a transcendental equation (inset, see [74] for details). The mean completion time of a gated reaction. —To pro-ceed, we recall that here we assumed Markovian internal dy-namics which means that W NR is exponentially distributedwith rate β . We will now also assume that W is exponen-tially distributed with rate γ , but note that this assumption ismade for clarity and that it can be easily generalized. Also,note that waiting times in other sites are kept general. Takingexpectations in Eq. (4) we find [74] h T ( NR ) i = γ − + h T FP ( X ) i K D + π R h − ˜ T FP ( X , λ ) i , (5)where K D = β / γ , π R = β / λ , and with λ = α + β . Equa-tion (5) asserts that h T ( NR ) i can be determined provided themean and distribution of the ungated return time T FP ( X ) .To better understand Eq. (5), we observe that the h T ( NR ) i ∼ β − asymptotics holds for both large and smallvalue of β . Specifically, when β (cid:29) T ( NR ) in Eqs. (1) and (2) concludingthat the gated and ungated problems are practically equivalent.In the other extreme, β (cid:28)
1, the transition to the non-reactivestate becomes rate limiting and h T ( NR ) i (cid:29) γ (cid:29) h T ( NR ) i = π − h − ˜ T FP ( X , λ ) i − h T FP ( X ) i . Tounderstand this result observe that in this limit the probabil-ity that the particles returns to the origin in the reactive stateis given by p = π R h − ˜ T FP ( X , λ ) i . Thus, on average, theparticle returns to the origin 1 / p times before a reaction oc-curs with each return taking h T FP ( X ) i time units on average. In the other extreme, γ (cid:28)
1, the particle is slow to leave theorigin and a reaction occurs following a transition to the non-reactive state. In this limit we find h T ( NR ) i = / β .The mean reaction time in Eq. (5) scales linearly withthe mean of T FP ( X ) as expected. To better understand howfluctuations in the return time to the origin affect the result,we neglect the dissociation-time γ − and expand ˜ T FP ( X , λ ) by its moments to second order in λ . Doing this we obtain h T ( NR ) i ’ β − + π − ( + CV ) h T FP ( X ) i , where CV = σ ( T FP ( X )) / h T FP ( X ) i stands for the coefficient of variation[74]. We thus see that fluctuations in T FP ( X ) tend to increasethe mean completion time of the gated reaction. In fact, in-voking Jensen’s inequality we get h T ( NR ) i ≥ γ − + h T FP ( X ) i K D + π R h − e − λ h T FP ( X ) i i (6)where the equality holds if and only if σ ( T FP ( X )) =
0. Adeterministic return time thus yields a universal lower boundfor h T ( NR ) i , and any fluctuation around the mean return time h T FP ( X ) i will necessarily slow down the completion of thegated reaction. This effect is illustrated in Fig. 3. Beyond the mean. —We now turn attention to the distribu-tion of T ( NR ) . Laplace transforming Eq. (4) we get [74]˜ T ( NR , s ) = π − K D + ˜ T FP ( X , s ) − ˜ T FP ( X , s + λ ) π − ( s γ + K D + ) − K eq ˜ T FP ( X , s ) − ˜ T FP ( X , s + λ ) , (7)where K eq = α / β ; and it can once again be appreciated thatthe gated reaction time can be put in terms of the ungatedreturn time to the origin T FP ( X ) .Equation (7) is indispensable in cases where h T FP ( X ) i diverges. Equation (5) then provides little information, butEq. (7) can still be used to e.g., show that T ( NR ) inheritsthe tail asymptotics of T FP ( X ) . Specifically we find that if˜ T FP ( X , s ) ’ − ( τ s ) θ for s (cid:28)
1, with 0 < θ < τ > T ( NR , s ) ’ − π − − ˜ T FP ( X , λ ) + λ / γ ( τ s ) θ . (8)From here it follows that if the survival function of T FP ( X ) decays as ∼ t − θ , then so does that of T ( NR ) ; and the prefac-tor can be determined by Eq. (8) and the Tauberian theorem. Putting it all to work. —To illustrate the applicability of ourapproach, consider now a simple symmetric random walk ona 1d lattice. Starting the walk at r , and working in discretetime, it is well known that the Z-transform of the first-passagetime to the origin is given by ˆ T FP ( r , z ) = ( − √ − z z ) | r | [14].The corresponding solution in continuous time is then givenby ˜ T FP ( r , s ) = ( − √ − ˜ ψ ( s ) ˜ ψ ( s ) ) | r | , where we have simply re-placed z with the Laplace transform of the waiting-time distri-bution, ˜ ψ ( s ) , in the CTRW [75]. Finally, we trivially observethat when such a symmetric random walk leaves the origin itwill be found at ± T FP ( X , s ) = ˜ T FP (+ , s ) . Substituting these results into Eq.(7) and then (3), the solution to the corresponding gated prob-lem is readily obtained; and we have verified that this solutionidentifies with the solution obtained by Budde, C´aceres andR´e for a particle that is initially prepared in the reactive state[41, 42]. In lieu of the general approach developed herein, thelatter was obtained with admirable effort.Continuing with the same example, we plot numerical in-versions [76] of the reaction time distribution in Eq. (3) (Fig.4). Two gated cases, and the corresponding ungated case, arecompared. We first observe that the power-law governing thelong time asymptotics of all distributions is identical to the ∼ t − / decay which is characteristic to the first-passage prob-ability of the 1d random walk. These asymptotic results agreewith the prediction of Eq. (8). However, the behaviour at in-termediate times differs significantly between gated cases. Forthe first case (orange), we take β = β = − and find that this leads to theemergence of a wide intermediate time window that is gov-erned by a ∼ t − / power-law decay. This “cryptic regime”was also observed in the analogous diffusion problem studiedby Mercado-V´asquez and Boyer [55], but here we see that itcan also be accompanied by an additional exotic and previ-ously unobserved kinetic effect which renders the gate reac-tion time distribution multimodal (Fig. 4, inset). We trace thiseffect to the existence of two populations of particles: thosewhich reached the origin without ever switching to the non-reactive state and those which switched prior to reaching. Asthe latter are blocked from reacting for a mean time 1 / β (cid:29) FIG. 4. Gated reaction time distributions for CTRW on a 1d lat-tice and the ungated behaviour for comparison. Here, the particle istaken to start from position r = + α = β = − (red) and β = γ =
1. Solid lines come fromnumerical inversion of Eq. (3) and full circles from Monte-Carlosimulations. that is much larger than the median return time to the origin,two distinct peaks are formed.
Numerical applications. —The approach taken for the 1dCTRW extends to many other first-passage time problems forwhich analytical results are known. In all such cases, theframework developed herein can be used to readily yield cor-responding solutions for gated reaction times. However, inmany complicated scenarios analytical results are not avail-able and one must then resort to numerical simulations. Wenow explain how these can be considerably sped up by tak-ing advantage of the relations derived above. Consider forexample the mean time of a gated reaction, and imagine thatthe latter should be determined via numerical simulations fora wide range of rate constants α , β and γ . While this processcan be extremely time consuming, one can instead take advan-tage of Eq. (5) which only requires numerical determinationof the mean and distribution of the ungated first-passage time T FP ( X ) . As the latter does not depend on the above param-eters it can be determined once and for all, thus providing aquick and efficient way of getting gated reaction times. Simi-lar reasoning applies to the mean gated reaction time in Eq.(2), which in turn requires determination of one additionalungated first-passage time: T FP ( r ) . In [74] we exemplifysuccessful implementation of this procedure for the gated re-action depicted in Fig. 1. Conclusions. —In this letter we developed a unified ap-proach to gated reactions on networks, and the results ob-tained were used to extract considerable insight. Applicationscover the entire spectrum of chemical reactions and are espe-cially relevant to the emerging field of single-molecule chem-istry. As our framework extends current knowledge on thelong studied topic of first-passage it also applies more broadly,and shall be particularly useful in the context of search andforaging where similar ideas also apply.
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Yuval Scher ∗ and Shlomi Reuveni School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry,and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n CONTENTS
SI. Derivation of Eq. (2) in the main text 2SII. Derivation of Eq. (3) in the main text 2SIII. Derivation of Eq. (5) in the main text 3SIV. Small λ expansion of h T ( NR ) i SI. DERIVATION OF EQ. (2) IN THE MAIN TEXT
Taking expectations of both sides of Eq. (1) in the main text we get h T ( x ) i = h T FP ( r ) i + h I FP i h T ( NR ) i (S1)where we have used the independence of I FP (the internal state at the moment of the first-passage time to the origin) and T ( NR ) (the gated reaction time when starting from NR ≡ ( , NR ) ) to write: h I FP T ( NR ) i = h I FP i h T ( NR ) i . Indeed, the latter followsfrom the fact that the gated reaction time starting at the origin can only depend on I FP via the internal state, which in T ( NR ) hasalready been set to be non-reactive. We are then left with the task of calculating h I FP i which is given by h I FP i = · h P ( R , T FP ( r ) | q ) i + · h P ( NR , T FP ( r ) | q ) i = h P ( NR , T FP ( r ) | q ) i . (S2)Explicitly, when q = NR, we have and P ( NR , t | NR ) = π NR + π R e − λ t , and so h P ( NR , T FP ( r ) | NR ) i = π NR + π R ˜ T FP ( r , λ ) , (S3)where ˜ T FP ( r , λ ) is the Laplace transform of T FP ( r ) evaluated at λ . Similarly, when q = R, we have P ( NR , t | R ) = π NR ( − e − λ t ) , and so h P ( NR , T FP ( r ) | R ) i = π NR − π NR ˜ T FP ( r , λ ) . (S4)Equation (2) in the main text follows directly from Eqs. (S1)-(S4) above. SII. DERIVATION OF EQ. (3) IN THE MAIN TEXT
Here again, we start from Eq. (1) in the main text. Laplace transforming it we get˜ T ( x , s ) = D e − sT ( x ) E = * e − s h T FP ( r )+ I FP T ( NR ) i+ , (S5)which in turn gives ˜ T ( x , s ) = D P ( R , T FP ( r ) | q ) e − sT FP ( r ) E + * P ( NR , T FP ( r ) | q ) e − s h T FP ( r )+ T ( NR ) i+ = D P ( R , T FP ( r ) | q ) e − sT FP ( r ) E + D P ( NR , T FP ( r ) | q ) e − sT FP ( r ) E D e − sT ( NR ) E . (S6)Setting q = NR and using P ( R , t | NR ) = π R ( − e − λ t ) and P ( NR , t | NR ) = π NR + π R e − λ t , we get˜ T ( x , s ) = D π R e − sT FP ( r ) − π R e − ( s + λ ) T FP ( r ) E + D π NR e − sT FP ( r ) + π R e − ( s + λ ) T FP ( r ) E D e − sT ( NR ) E = π R ˜ T FP ( r , s ) + π NR ˜ T FP ( r , s ) ˜ T ( NR , s ) − π R ˜ T FP ( r , s + λ ) + π R ˜ T ( NR , s ) ˜ T FP ( r , s + λ ) , (S7)Similarly, setting q = R and using P ( R , t | R ) = π R + π NR e − λ t and P ( NR , t | R ) = π NR (cid:16) − e − λ t (cid:17) , we get˜ T ( x , s ) = D π R e − sT FP ( r ) + π NR e − ( s + λ ) T FP ( r ) E + D π NR e − sT FP ( r ) − π NR e − ( s + λ ) T FP ( r ) E D e − sT ( NR ) E = π R ˜ T FP ( r , s ) + π NR ˜ T FP ( r , s ) ˜ T ( NR , s ) + π NR ˜ T FP ( r , s + λ ) − π NR ˜ T ( NR , s ) ˜ T FP ( r , s + λ ) , (S8)Equation (3) in the main text follows directly from Eqs. (S5)-(S8) above. SIII. DERIVATION OF EQ. (5) IN THE MAIN TEXT
To derive Eq. (5) in the main text, we start from Eq. (4) in the main text and recall that the internal dynamics was assumed to beMarkovian. Thus, the probability density function of the waiting time in the non-reactive state is given by φ ( t ) = β e − β t , and thecumulative distribution function is correspondingly given by Φ ( t ) = P ( W NR < t ) = − e − β t . To proceed, we also assume that thewaiting time of the particle at the origin is taken from an exponential distribution with probability density function ψ ( t ) = γ e − γ t ,and cumulative distribution function Ψ ( t ) = P ( W < t ) = − e − γ t .We now use Eq. (4) to write the left hand side of Eq. (5) as h T ( NR ) i = P ( W NR < W ) h W NR | W NR < W i + P ( W NR ≥ W ) h W + T FP ( X ) + I FP T ( NR ) | W NR ≥ W i : = + . (S9)To obtain the desired result, we start by evaluating 1 as1 = P ( W NR < W ) h W NR | W NR < W i = P ( W NR < W ) R ∞ t φ ( t )( − Ψ ( t )) dtP ( W NR < W ) = Z ∞ t φ ( t )( − Ψ ( t )) dt = β ( β + γ ) . (S10)We then move on to calculate the second, more challenging, term 2 . Utilizing the the fact that T FP ( X ) , I FP and T ( NR ) are independent of W and W NR , we obtain2 = P ( W NR ≥ W ) h W | W NR ≥ W i + P ( W NR ≥ W ) h h T FP ( X ) i + (cid:10) I FP T ( NR ) (cid:11) i : = A + B . (S11)Calculation of A is done in the same manner as in Eq. (S10) to give γ / ( β + γ ) . For B we note that P ( W NR ≥ W ) = γ / ( β + γ ) and recall that h I FP T ( NR ) i = h I FP i h T ( NR ) i from independence. We then use the fact that this is a renewal process, hence T ( NR ) and T ( NR ) are IID and in particular: h T ( NR ) i = h T ( NR ) i . To get h I FP i we use Eq. (S3) where we recall that here q = NR by construction. Plugging everything back together and rearranging we obtain Eq. (5) in the main text h T ( NR ) i = γ + h T FP ( X ) i K D + π R h − ˜ T FP ( X , λ ) i , (S12)with K D = βγ . SIV. SMALL λ EXPANSION OF h T ( NR ) i Taking the limit γ → ∞ in Eq. (5), and expanding ˜ T FP ( X , λ ) by its moments to second order in λ , we get h T ( NR ) i = π − h T FP ( X ) i λ h T FP ( X ) i − λ h T FP ( X ) i . (S13)Rearranging, h T ( NR ) i = π − λ − λ h T FP ( X ) ih T FP ( X ) i = β − − λ h T FP ( X ) ih T FP ( X ) i . (S14)By definition of the variance σ ( T FP ( X )) = h T FP ( X ) i − h T FP ( X ) i , so we can rewrite Eq. (S14) as h T ( NR ) i = β − − λ (cid:16) h T FP ( X ) i + σ h T FP ( X ) i (cid:17) . (S15)Recalling that + x ≈ − x for x (cid:28)
1, we have h T ( NR ) i = β − h + λ (cid:16) h T FP ( X ) i + σ h T FP ( X ) i + . . . (cid:17)i = β − + π − ( + CV ) h T FP ( X ) i + . . . , (S16)wherewe have used λβ = π − and the definition of the coefficient of variation CV = σ ( T FP ( X )) h T FP ( X ) i . SV. GAMMA-DISTRIBUTED RETURN TIMES
In Fig. 3 of the main text we consider a return time T FP ( X ) that comes from a Gamma distribution with shape parameter n andscale parameter θ . The probability density function of this return time is then given by f ( t ) = Γ ( n ) θ n t n − e − t θ , (S17)which gives a mean of h T FP ( X ) i = n θ , and the following Laplace transform h e − sT FP ( X ) i = ( + s θ ) − n . (S18)Inserting these expressions into Eq. (5) in the main text, while setting α = β = λ , yields the mean gated reaction time h T ( NR ) i = ( n θ γ + ) λ + γ − γ ( + θ λ ) − n . (S19)To study the effect that fluctuations in the return time have on h T ( ) i , we set θ = / n such that the mean of T FP ( X ) is set to 1while its coefficient of variation is given by CV = / √ n . In this way, we can tune CV by changing the shape parameter n , whilekeeping the mean of the return time distribution fixed.In Fig. 3a of the mean text we plot the mean gated reaction time in Eq. (S19) for different values of CV . Note that the limitof CV →
0, i.e., the case of deterministic return times, is obtained by taking n → ∞ , which giveslim CV → h T ( NR ) i = e λ ( + γ ) e λ ( λ + γ ) − γ . (S20)As we show in the main text, the expression in Eq. (S20) serves as a lower bound of the mean reaction time in Eq. (S19).In Fig. 3b, we plot the mean reaction time normalized by its lower bound in Eq. (S20). We demonstrate that the deviation fromthe lower bound is CV dependent, and that for each CV value there is a corresponding λ ∗ for which this deviation is maximal.To find this value we look for extrema of the normalized gated reaction time. We found that every extremum must satisfy thefollowing transcendental equation: e λ ∗ = ( n + λ ∗ ) ( + γ + λ ∗ ) (cid:16) + λ ∗ n (cid:17) n − γλ ∗ λ ∗ + n ( + γ + λ ∗ ) , (S21)which can be solved numerically to obtain λ ∗ for any given value of CV = / √ n . SVI. DERIVATION OF EQ. 7 IN THE MAIN TEXT
To derive Eq. (7) in the main text, we start from Eq. (4) in the main text and once again recall that the internal dynamicswas assumed to be Markovian. Thus, the probability density function of the waiting time in the non-reactive state is given by φ ( t ) = β e − β t , and the cumulative distribution function is correspondingly given by Φ ( t ) = P ( W NR < t ) = − e − β t . To proceed,we also assume that the waiting time of the particle at the origin is taken from an exponential distribution with probability densityfunction ψ ( t ) = γ e − γ t , and cumulative distribution function Ψ ( t ) = P ( W < t ) = − e − γ t .Laplace transforming Eq. 4 in the main text we obtain˜ T ( NR , s ) = h e − sT ( NR ) i = P ( W NR < W ) h e − sW NR | W NR < W i + P ( W NR ≥ W ) h e − s h W + T FP ( X )+ I FP T ( NR ) i | W NR ≥ W i : = + . (S22)We start by evaluating 1 , which can be written explicitly as1 = P ( W NR < W ) h e − sW NR | W NR < W i = P ( W NR < W ) L { φ ( t )( − Ψ ( t )) } P ( W NR < W ) = L { φ ( t )( − Ψ ( t )) } = β s + β + γ . (S23)We then move on to calculate the second, more challenging, term 2 . Utilizing the the fact that T FP ( X ) , I FP and T ( NR ) are independent of W and W NR , we obtain2 = P ( W NR ≥ W ) h e − sW | W NR ≥ W i h e − s h T FP ( X )+ I FP T ( NR ) i i . (S24)The first two terms being multiplied can be calculated in the same manner as in Eq. (S23) above to give L { ψ ( t )( − Φ ( t )) } = γ s + β + γ . The third term can be calculated as we did in Eq. (S6), to give2 = L { ψ ( t )( − Φ ( t )) } h h P ( R , T FP ( X ) | NR ) e − sT FP ( X ) i + h P ( NR , T FP ( X ) | NR ) e − sT FP ( X ) i h e T ( NR ) i i . (S25)To evaluate the above we recall that P ( R , t | NR ) = π R ( − e − λ t ) and P ( NR , t | NR ) = π NR + π R e − λ t ; and further use the fact that T ( NR ) is an IID copy of T ( NR ) . The overall expression for the Laplace transform of the gated reaction time T ( NR ) can thenbe written as ˜ T ( NR , s ) = h e − sT ( NR ) i = β s + β + γ + γ s + β + γ h π R h e − sT FP ( X ) i − π R h e − ( s + λ ) T FP ( X ) i i + γ s + β + γ h π NR h e − sT FP ( X ) i + π R h e − ( s + λ ) T FP ( X ) i i h e − sT ( NR ) i . (S26)Rearranging to solve for h e − sT ( ) i , we obtain˜ T ( NR , s ) = h e − sT ( ) i = β s + β + γ + γ s + β + γ h π R h e − sT FP ( X ) i − π R h e − ( s + λ ) T FP ( X ) i i − γ s + β + γ h π NR h e − sT FP ( X ) i + π R h e − ( s + λ ) T FP ( X ) i i . (S27)Multiplying both numerator and denominator by s + β + γπ R γ , we get Eq. (7) in the main text˜ T ( NR , s ) = βπ R γ + h e − sT FP ( X ) i − h e − ( s + λ ) T FP ( X ) i s + β + γπ R γ − K eq h e − sT FP ( X ) i − h e − ( s + λ ) T FP ( X ) i : = π − K D + h e − sT FP ( X ) i − h e − ( s + λ ) T FP ( X ) i π − ( s γ + K D + ) − K eq h e − sT FP ( X ) i − h e − ( s + λ ) T FP ( X ) i . (S28) SVII. DERIVATION OF EQ. 8 IN THE MAIN TEXT
To derive Eq. (8) in the main text we plug into Eq. (7) ˜ T FP ( X , s ) ’ − ( τ s ) θ for s (cid:28)
1, with 0 < θ < τ >
0. This gives˜ T ( NR , s ) ’ π − K D + h − ( τ s ) θ i − ˜ T FP ( X , s + λ ) π − ( s γ + K D + ) − K eq h − ( τ s ) θ i − ˜ T FP ( X , s + λ ) , (S29)which can be simplified by algebraic manipulations as follows˜ T ( NR , s ) ’ h π − K D + − ˜ T FP ( X , s + λ ) ih − ( τ s ) θ π − K D + − ˜ T FP ( X , s + λ )] ih π − ( K D + ) − K eq − ˜ T FP ( X , s + λ ) ih + K eq ( τ s ) θ + π − s γ π − ( K D + ) − K eq − ˜ T FP ( X , s + λ ) i . (S30)Noting that π − − K eq = α + ββ − αβ =
1, we cancel matching terms and in numerator and denominator to obtain˜ T ( NR , s ) ’ h − ( τ s ) θ π − K D + − ˜ T FP ( X , s + λ ) i + K eq ( τ s ) θ + π − s γ π − ( K D + ) − K eq − ˜ T FP ( X , s + λ ) . (S31)In the limit s → π − s γ with respect to K eq ( τ s ) θ , and further approximate ˜ T FP ( X , s + λ ) ’ ˜ T FP ( X , λ ) . Then,expanding the fraction on the right, we obtain˜ T ( NR , s ) ’ h − ( τ s ) θ π − K D + − ˜ T FP ( X , λ ) ih − K eq ( τ s ) θ π − ( K D + ) − K eq − ˜ T FP ( X , λ ) i . (S32)Now by carefully multiplying these terms, neglecting higher order products and simplifying the result, we get Eq. (8) of themain text. SVIII. ANALYSING THE GATED REACTION DEPICTED IN FIG. 1 OF THE MAIN TEXT
FIG. S1. The mean completion time of the gated reaction depicted in Fig. 1 of the main text vs λ . Results from direct Monte-Carlosimulations for a handful of λ values (circles) are compared to the hybrid method in which Monte-Carlo simulations are only used to evaluatethe distributions of the ungated first-passage times T FP ( X ) and T FP ( r ) . These are then numerically Laplace transformed and fed into Eq.(5) which consequently feeds into Eq. (2) to determine the mean gated reaction time for all values of λ (blue line). The hybrid method opensthe door for exhaustive scan of governing parameters in complex gated reactions. Such exhaustive scans are virtually impossible with directMonto-Carlo simulations as these are much slower in comparison. Lastly, we demonstrate how the framework developed in our letter can be used to speed up and greatly facilitate numericalstudies of gated reactions on networks. To this end, we consider the gated reaction depicted in Fig. 1 of the main text. Taking thesame spatial initial condition described in the figure, we set equilibrium initial conditions with π NR = π R = / λ = α = β ). For the waiting time of the particle in the different sites we have taken the exponentialdistribution with mean 1, and further assumed that the particle jumps only to nearest neighbours and with equal probabilities.The mean reaction time was then obtained for a handful of λ values by direct Monte-Carlo simulations (Fig. S1, circles). Notethat for each value of λ we performed 10 simulations which serves to illustrate that exhaustive scan of the parameters space isimpractical and computationally costly. In contrast, the mean reaction time can be readily computed using the hybrid methodthat we described in the main text. In this method, Monte-Carlo simulations are only used twice to estimate the ungated reactiontimes T FP ( X ) and T FP ( r ) . This information is then used as input which feeds into Eqs. (5) and (2) to determine the meancompletion time of the gated reaction for all values of λλ