Nonlocality of relaxation rates in disordered landscapes
Yunyun Li, Debajyoti Debnath, Pulak K. Ghosh, Fabio Marchesoni
NNonlocality of relaxation rates in disordered landscapes
Yunyun Li,
1, 2, 3, a) Debajyoti Debnath, Pulak K. Ghosh, and Fabio Marchesoni
1, 2, 5 Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University,Shanghai 200092, People’s Republic of China China-EU Joint Lab for Nanophononics, Tongji University, Shanghai 200092,China Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology,School of Physics Science and Engineering, Tongji University, Shanghai 200092,China Department of Chemistry, Presidency University, Kolkata 700073, India Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy (Dated: November 12, 2018)
We investigate both analytically and by numerical simulation the relaxation of an overdamped Brownianparticle in a 1D multiwell potential. We show that the mean relaxation time from an injection point insidethe well down to its bottom is dominated by statistically rare trajectories that sample the potential profileoutside the well. As a consequence, also the hopping time between two degenerate wells can depend onthe detailed multiwell structure of the entire potential. The nonlocal nature of the transitions between twostates of a disordered landscape is important for the correct interpretation of the relaxation rates in complexchemical-physical systems, measured either through numerical simulations or experimental techniques.
I. INTRODUCTION
The problem of time relaxation around a local mini-mum of a free-energy landscape is ubiquitous in chem-ical physics. In fact, the landscape picture assumes anatural separation of low-temperature molecular motionsampling distinct potential energy minima, and vibrationwithin a minimum. The manner in which a disorderedmaterial samples its landscape as a function of tempera-ture thus provides information on its long-time relaxationproperties. The energy landscape paradigm has beensuccessfully applied to protein folding , the mechanicalproperties of glasses , and the dynamics of supercooledliquids .In this context, Adam-Gibbs’ formula suggests aphenomenological connection between kinetics and ther-modynamics in disordered systems, that is, τ = A exp( B/T s c ), where τ is a relaxation time, A and B are two phenomenological constants, and s c is a configu-rational entropy factor related to the number of minimaof the system’s multidimensional energy surface. For in-stance, at low enough temperatures the system becomesstuck in a single minimum, the depth of which increasesas the cooling rate decreases: this describes a glass tran-sition. In this context, of prominent interest is the caseof relaxation between two degenerate free-energy minimaseparated by an (almost) symmetric activation barrier.In the current literature this is referred to as the Kramers’problem . In calculating the average transition time be-tween two such states, one typically ignores the presenceof other possible less stable (more energetic) states inthe free-energy landscape . We show that statisticallyrare trajectories that connect two such degenerate states a) Electronic mail: [email protected] only after entering another neighboring state, are respon-sible for an increase of the relevant mean transition time,sometimes by orders of magnitude. The consequence isthat in order to ignore the contribution of slowly mean-dering trajectories and keep using the results of standardKramers’ theory, one has to restrict the system’s phase-space volume defining the free-energy stable states.Our conclusion has an immediate counterpart and,hopefully, application in the strategies of path samplingfor the numerical investigation of complex systems .For instance, an unfolded protein can explore thousandsof intermediate structures (conformations) before reach-ing a long-lived (stable) folded conformation. The mostnumerically efficient approach to investigate this processinvolves simulating protein folding with molecular dy-namics for a relatively short time, and then analyzing theresulting trajectories to extract a coarse-grained Markovstate model (MSM). An MSM consists of an appropriatechoice of long-lived clustered conformational states andthe transition rates between them. To create an MSM,one runs molecular dynamics simulations to determinehow frequently a protein changes from one state to an-other, and clusters intermediate structures based on ki-netic proximity (e.g., how energetically easy is switchingfrom one structure to another). The transition rates aretypically determined by averaging the time the proteintakes to switch between any two states encoded in theMSM. Due to the coarse-grained nature of the MSM, acontinuous trajectory connecting a pair of sampled statesmight well enter first the phase-space basin belonging toanother state without being trapped there. This occur-rence, though unlikely, may dramatically affect the corre-sponding transition time. How to correctly generate thereactive trajectories representing a specific transition ofinterest for the MSM is an issue of ongoing research.The contents of this paper is organized as follows. InSec. II we first simulate the relaxation of an overdamped a r X i v : . [ phy s i c s . c h e m - ph ] F e b Brownian particle in a 1D potential well. We determineboth numerically and analytically the mean first-passagetime (MFPT) for the particle to reach the bottom ofthe well from an injection point inside it. We show thatwhen the injection point rests inside the well, but higherthan the bottom of another adjacent well, then the raretrajectories crossing the barrier separating the two wellsbecome dominant and, on lowering the noise level, theMFPT increases exponentially. Some of the results pre-sented here have been independently derived in Ref. for discrete stochastic models of biological interest. InSec. III we interpret this effect by distinguishing be-tween two types of trajectories, the most probable tra-jectories pointing from the injection point straight downto the well bottom, and the rare trajectories overcomingthe barrier into the side well. The distribution densityof the relaxation times allows a clear-cut distinction be-tween these two types of trajectories. In Sec. IV weextend our analysis to the case of multiwell potentialsand conclude that the MFPT inside a well is dominatedby barrier-crossing anytime the particle’s injection pointrests above the level of the lowest lying among all adja-cent wells (Sec. IV A). Finally, we consider the case ofthe hopping process between two degenerate minima ofthe potential and discuss how the MFPT over the barrierseparating them can depend on the level of the injectionpoint and, therefore, on the multistable structure of theentire potential (Sec. IV B). In Sec. V we draw someconcluding remarks regarding the impact of this effecton the interpretation of actual relaxation measurements. II. RELAXATION TIMES IN A BISTABLE POTENTIAL
We start introducing two categories of trajectories a1D system may take while relaxing toward a stable state.Broadly speaking, we distinguish between regular trajec-tories, the most probable and typically the shortest ones,given certain initial conditions, and a subset of domi-nant trajectories, which one determines with referenceto the observable being measured. The most probabletransition trajectories in a 1D system has been classifiedby analyzing the (local) minima of the relevant actionintegrals . Here, we are rather concerned with iden-tifying the systems’ trajectories that most contribute tothe mean value of a specific observable of interest.A study-case is represented by the transition times t ( a, x ) of an overdamped Brownian particle obeying theLangevin equation (LE),˙ x = − V (cid:48) ( x ) + ξ ( t ) , (2.1)where x ( t ) denotes the particle coordinate, V ( x ) is a con-fining multistable potential, and ξ ( t ) models a stationary,zero-mean, Gaussian noise source with autocorrelationfunction (cid:104) ξ ( t ) ξ (0) (cid:105) = 2 Dδ ( t ) . (2.2) -1 0 110 -1 1 -0.40.4 a .. . T x V(x) . x s b c x V(x)
Figure 1. (Color online) Mean first-passage time T = T ( a, x )vs. x from numerical integration of the LE (2.1) with theasymmetric bistable potential V ( x ) = x / − x + x/ D . The potential minima arelocated at x a ≈ − .
088 and x c ≈ . x b ≈ . V ( x s ) = V c , at x s ≈ − . T ( a, x ) were obtained byperforming the double integral in Eq. (2.4) for the appropriate D . The particle will be injected at a given point x and takenout upon reaching the exit point x a . To keep our nota-tion as simple as possible, we place the exit point at thebottom of a potential well, termed well a , located on theleft of the injection point, i.e., x a < x , see inset of Fig.1. The time length of each trajectory is the observableof interest, t ( a, x ).The average transition time T ( a, x ) ≡ (cid:104) t ( a, x ) (cid:105) forthe particle to diffuse from x to a , is given by the well-known MFPT formula , T ( a, x ) = 1 D (cid:90) x x a dyp ( y ) (cid:90) ∞ y p ( z ) dz, (2.3)where p ( x ) = N exp[ − V ( x ) /D ] is the stationary proba-bility density of the process (2.1). Note that for a con-fining potential, lim x →±∞ p ( x ) = 0, i.e., x → ∞ can betreated as a reflecting boundary We specialize now Eq. (2.3) to the case of an asym-metric bistable potential. As illustrated in the inset ofFig. 1, x b locates the top of the barrier, b , and x a and x c denote the bottom of the left, a , and right well, c ,respectively, with V a < V c . Here and in the follow-ing, we adopted the short-hand notation V ( x a ) = V a , V ( x b ) = V b , V ( x c ) = V c , V ( x ) = V , and prime for an x derivative, ( . . . ) (cid:48) = d ( . . . ) /dx . The threshold x s is thepoint on the r.h.s. of a that has the same potential energyas the bottom of well c ; for the asymmetric double-wellpotential of Fig. 1, V ( x s ) = V c with x s > x a .We then estimate the MFPT (2.3) in the weak noiselimit, D < V b − V c , for three different ranges of the injec-tion point, x : (i) out-of-well, x > x b . The functions p ( x ) and p − ( x )are sharply peaked, respectively, around points x a and x c and around point x b . As a consequence, for x > x b thenested integrals (2.3) factorize, that is, T ( a, x ) = 1 D (cid:90) x x a dyp ( y ) (cid:90) ∞ x b p ( z ) dz. (2.4)In the limit of weak noise p ( z ) /p ( y ) (cid:39) exp[( V b − V c ) /D −| V (cid:48)(cid:48) b | ( x − x b ) / D − V (cid:48)(cid:48) c ( x − x c ) / D ], so that the integrals(2.4) can be approximated to T ( a, x ) = 2 π (cid:112) | V (cid:48)(cid:48) b | V (cid:48)(cid:48) c exp (cid:18) V b − V c D (cid:19) . (2.5)This is the well-known Kramers’ formula, T K ( a, c ), forthe escape time out of well c . Here, according to our no-tation, all escape trajectories are regular and the ensuing(almost x independent) relaxation time is characterizedby the slow relaxation process x c → x a . (ii) barrier well region, x s < x < x b . For this choiceof the injection point, the first integrand (2.4) can beapproximated to p − ( y ) (cid:39) exp[ V /D + V (cid:48) ( y − x ) /D ];hence T ( a, x ) = 1 | V (cid:48) | (cid:115) πDV (cid:48)(cid:48) c exp (cid:18) V − V c D (cid:19) . (2.6)Here we took the absolute value of V (cid:48) only for the sakeof generality. This result is suggestive: Although theparticle was injected directly in well a , still it takes anexponentially long average time to reach its bottom, x a .Moreover, in contrast with Kramers’ time of Eq. (2.5), T ( a, x ) appears to depend on how high the injectionpoint lies with respect to the minimum, V c , of the side-well c . As discussed in Sec. III, the MFPT (2.6) is indeeddominated by the rare trajectories that cross over intowell c before being absorbed at x a . (iii) bottom well region, x < x s . As x approachesthe exit point, one can easily take the x → x a limit ofthe double integral (2.3), thus obtaining the logarithmiclaw, T ( a, x ) = 12 V (cid:48)(cid:48) a (cid:20) γ + ln (cid:18) V − V a D (cid:19)(cid:21) , (2.7)where γ (cid:39) .
577 is the Mascheroni’s constant. This isthe short MFPT one would expect on account of the soleregular trajectories of the relaxation process. Indeed,such trajectories run straight downhill from x subject toweak noise fluctuations, whose effect grows appreciableonly close to the exit point, x = x a .Our analytical estimates (2.5)-(2.7) reproduce well thethree different regimes of the T ( a, x ) curves of Fig. 1,obtained by numerically computing the double integral(2.3) for very small D values. The crossover betweenthe logarithmic (2.7) and the exponential branch (2.6) of T ( a, x ) is fairly sharp, because the exponential in Eq.(2.6) abruptly vanishes for x < x s and | V − V c | (cid:28) D .In passing we notice that our approximations (2.5) and(2.6) coincide (apart from minor typographical errors)with the first two MFPT’s reported in Eq. (33) of Ref. -9 -7 x = ; -0.15; x = x s x = P ( t ) t -0.15 -0.4 -0.6 -0.8 P ( t ) t (x 10 ) Figure 2. (Color online) Distribution densities, P ( t ), of thetransient times t = t ( a, x ) obtained by numerically integrat-ing the LE (2.1) for the asymmetric bistable potential of Fig.1 with D = 0 .
01 and different x . The dashed curves repre-sent the harmonic approximation P s ( t ) of Eq. (3.5) for thetwo x closest to x a , see text. Inset: semi-logarithmic plot of P ( t ) vs t for three values of x and D = 0 .
01. The three datasets are closely fitted by the function P l ( t ) in Eq. (3.4). Thedashed line with T K ( x a , x ) has been drawn to guide the eye. for Schl¨ogl’s model in the large size system limit. Ourderivation is much simpler, indeed, but restricted to thecase of continuous stochastic transition processes.Finally, the results of this section can be readily ex-tended to the case when the side-well c is deeper than theexit well, V c < V a . Only approximation (2.7) needs to bemodified as the probability density, p ( x ), in the exit wellgets exponentially suppressed. As a consequence, theright hand side of Eq. (2.7) must be multiplied by the ad-ditional factor exp[( V a − V c ) /D ]. This means that, sinceno threshold x s could be defined, the average transitiontime is exponentially long for any in-well injection point,namely, T ( a, x ) ∝ exp[( V − V c ) /D ] for x a < x < x b . III. THE ROLE OF THE DOMINANT TRAJECTORIES
As anticipated in the foregoing section, the results ofEqs. (2.5) and (2.7) lend themselves to a simple interpre-tation in terms of regular trajectories. For x > x b theparticle is initially placed in the side-well c , so that it,first, relaxes around the local p ( x ) maximum at x c and,then, escapes into well a by overcoming the barrier b ; asa consequence T ( a, x ) is quite insensitive to the injec-tion point x . For x a < x < x s the particle tends to rolldownhill toward the exit point x a , corresponding to theabsolute maximum of p ( x ), with a short average tran-sition time proportional to the logarithm of the initialdisplacement, x − x a .The transitions that start out in the barrier region x s < x < x b are qualitatively different. As the injectionpoint lies inside well a , the trajectories oriented towardthe exit point are still the most probable, or, stated oth-erwise, they represent the process’ regular trajectories,as expected. Nevertheless, the particle can diffuse from x over the barrier into well c with small but finite proba-bility. Following Refs. , we can estimate the splittingprobability π ( a, x ) for the particle to exit at a withoutfirst reaching c , and π ( c, x ) for the particle to fall intowell c before being absorbed at a , π ( c, x ) = 1 − π ( a, x ) , (3.1) π ( a, x ) = (cid:90) x c x dyp ( y ) / (cid:90) x c x a dyp ( y ) . (3.2)For weak noises and x not too close to the extrema x a and x b , the integral (3.2) can be approximated to π ( c, x ) (cid:39) | V (cid:48) | (cid:114) D | V (cid:48)(cid:48) b | π exp (cid:18) V − V b D (cid:19) . (3.3)Although the typical trajectories are by far the mostprobable – being π ( a, x ) (cid:39)
1, – still their contribution tothe average transition time T ( a, x ) is negligible, as theyreach the exit point in a quite short time, see Eq. (2.7).By contrast, the barrier crossings may well be very un-likely – being π ( c, x ) exponentially small, – but the par-ticle, after falling into well c , takes an exponentially longtime of the order of T K ( a, c ) [see Eq. (2.3) for x = c ],to recross into well a . The contribution to T ( a, x ) fromsuch rare trajectories amounts to π ( c, x ) T K ( a, c ), thatis, to our estimate in Eq. (2.6). In conclusion, as longas we characterize the relaxation in the overdamped po-tential V ( x ) by measuring the exit times, t ( a, x ), theotherwise sporadic trajectories crossing the barrier maybecome dominant , depending on the injection point. Ofcourse, this argument only applies for small, but finitenoise strengths, i.e., D → D = 0, there exists only one allowed determin-istic trajectory running downhill from x to x a for any a < x < x b .In real or numerical experiments one can easily samplerelaxation trajectories from x to x a and distribute themaccording to their temporal length, t ( a, x ). Based onthe argument above, where the regular trajectories areregarded as much faster than the dominant ones, the t -distribution density, P [ t ( a, x )], can be separated intotwo distinct terms, i.e., π ( a, x ) P s ( t ) + π ( c, x ) P l ( t ). Forthe sake of a comparison with actual data, in the regimeof weak noise one can introduce the approximations, P l ( t ) (cid:39) T ( a, x ) T K ( a, c ) exp[ − t/T K ( a, c )] , (3.4)for the long exit times of the statistically rare trajectoriescrossing the barrier, and P s ( t ) (cid:39) − √ π ddt (cid:20) V − V a D ( t ) (cid:21) · exp[ − ( V − V a ) /D ( t )] , (3.5) -1 0 110 x s1 s . s . . .. .. c x s2 T V(x) a (a) b c b x -1 0 110 T x D = V(x) x s2 . x s1 s . s . . ... c a b c b (b) Figure 3. (Color online) Mean first-passage time T = T ( a, x )vs. x in the asymmetric three-well potentials (a) V ( x ) = x / − x − . . x ) and (b) V ( x ) = x / − x − .
35 sin(6 x )for different D . Note that in (a) V c < V c and in (b) V c >V c ; in both cases V a < V c i , i = 1 ,
2. The data points arethe result of the numerical integration of LE (2.1) for therelevant choices of V ( x ) and D ; the dashed curves are thecorresponding analytical expressions of Eq. (2.4). with D ( t ) = e V (cid:48)(cid:48) a t −
1, for the intrawell relaxation tra-jectories. Our expression for P s ( t ) holds good for theharmonic approximation of the potential well a , that is,by setting V ( x ) = V a + (1 / V (cid:48)(cid:48) a ( x − x a ) and ignoringall anharmonic terms of the third order and higher. Itwas derived by standard MFPT methods and can bereformulated to match earlier solutions for t -distributionin a harmonic well . In Eq. (3.4) for P l ( t ), we approx-imated the probability of barrier crossing as π ( c, x ) (cid:39) T ( a, x ) /T K ( a, c ), and made use of the well-establishedexponential distribution for Kramers’ escape times from c back to a .In Fig. 2 we display the outcome of an extensive nu-merical simulation of the exit process, Eq. (2.1), for thepotential of Fig. 1 and different values of D . As theinjection point is shifted past the threshold x s , also therelaxation time distributions change abruptly. An ex-ponential tail associated with the dominant trajectoriesbecomes visible for x ≥ x s (inset); as predicted in Eq.(3.4), such a tail has a small amplitude of the order of T ( a, x ) /T K ( a, c ) and decays slowly with time constant T K ( a, c ). The distributions of the short relaxation timesdue to the regular trajectories, main panel, are reminis-cent of the t -distributions in a harmonic well, P s ( t ) ofEq. (3.5). However, the agreement gets quantitativelyclose only when x approaches x a , the convergence be-ing rather slow. We attributed this inconvenience to thespatial asymmetry of well a . Moreover, we remark thatthe average (cid:104) t ( a, x ) (cid:105) taken over the regular trajectoriesonly, namely by using the approximate distribution den-sity Eq. (3.5), is a monotonic decreasing function of D ;for vanishingly small D values it comes close to the pre-dicted estimate in Eq. (2.7). IV. GENERALIZATION TO MULTIWELL POTENTIALS
The results of Sec. II can be extended to study tran-sitions in multiwell potentials, as well. However, the al-gebraic manipulations on the MFPT (2.3) can becomemore complicated due to the multi-peaked structure ofthe functions p ( x ) and p − ( x ). Luckily, to gain a betterunderstanding of the role of the dominant trajectories inthe most general case of a disordered potential, it sufficesto analyze in some detail the three-well potentials, only.While any disordered potential can be regarded as an ap-propriate sequence of three-well potentials, it is clear thatthe relaxation properties discussed below only apply inthe limit of infinite observation times, where the diffusingparticle is allowed to explore the entire potential profile.Shorter observation times would necessarily restrict ouranalysis to the portion of the potential profile actuallyaccessed by the particle. A. Nondegenerate three-well potentials
Let us imagine to add a third well to the potentialplotted in Fig. 1. If we agree on that the exit well mustbe at the bottom of the lowest one, then two geometriesare possible, as illustrated in Fig. 3. Let c and c denote,respectively, the first and the second well to the right ofwell a , with barriers b and b separating the three wells.As for both wells V c i > V a , with i = 1 ,
2, the equations V ( x s i ) > V c i may define two thresholds, x s i , with a We consider now the special case of a three-well po-tential with two degenerate lower minima, say, in x a and x c , see Fig. 4. This means that wells a and c areequally deep, while the third well sits higher up, that is, V a = V c < V c . Then, the process (2.1) models the relax-ation occurring between two degenerate states, a mech-anism often invoked in the chemical physical literature.As discussed in Sec. I, for low noise levels this problemis commonly addressed by ignoring the presence of moreenergetic states in the neighborhood. However, the re-markable dependence of T ( x , a ) on the injection point, x , shown in figure, suggests a different picture. As longas x is confined around the bottom of well c , the MFPTfrom x to x a is almost independent of x and well repro-duced by the Kramers’ rate of Eq. (2.5) upon replacing x with c , and b with b . In this case the role of well c is irrelevant. However, on moving x to the right ofa certain threshold x s , T ( x , a ) suddenly jumps up to amuch higher value, insensitive to any further increase of x .The location of the threshold point s and the mag-nitude of the MFPT jump can be explained as fol-lows. We assume that the lower T ( x , a ) plateau for x b < x < x s is due to the regular trajectories cross-ing from c to a directly over barrier b and, there-fore, proportional to exp( V b − V c ), whereas the higherplateau must come from those rare trajectories that crossfirst barrier b to the right, with probability propor-tional to exp[ − ( V b − V )]. The time they take to crossback from well c to well c (and then to well a ) is aKramers’s time proportional to exp( V b − V c ). There-fore, their weighted contribution to the MFPT is pro-portional to exp( V − V c ) and, most remarkably, super-sedes the contribution from the regular trajectories for V − V c > V b − V c . Accordingly, x s is determined bychoosing V ( x s ) = V c + ∆ V , where ∆ V = V b − V c isthe barrier height separating wells a and c – see thegeometric construction in Fig. 4.As long as V ( x s ) < V b , the threshold x s is well de-fined. Therefore, there can exist a barrier region insidewell c , x s < x < x b , such that the relaxation trajec-tories creeping into well c are indeed dominant . Thecorresponding t -distributions are well fitted by doubleexponential functions (not shown) with decay constantsequal to the two plateau values of the curves T ( x , a )versus x . V. CONCLUSIONS Many systems in condensed matter are described byan overdamped particle that diffuses on a disordered en-ergy landscape of appropriate dimensionality, withoutever reaching a proper equilibrium state (glassy materialsare a good example). The physical chemical properties ofthese systems are often interpreted in terms of the relax-ation rates inside single locally stable states or betweenpairs of locally stable states. However, determining suchrates experimentally, through microscopic techniques, oreven numerically, may prove a moot problem. As dis-cussed in Secs. II and IV, the investigator who intendsto proceed by weakly exciting the system out of its lo-cally stable state and then letting it relax back to it, mayencounter the difficulty of establishing whether the mea-sured relaxation time depends on the presence of othermetastable states. This difficulty can be circumventedby a more restrictive definition of locally stable state.Our analysis clearly shows that in 1D the relaxationtimes within a single potential well or between degener-ate wells can be determined by ignoring additional po-tential wells only under the condition that the energy of what we call the injection point is sufficiently close to theenergy of the well bottom. How close, it depends on theactual distribution of the wells along the potential land-scape. Indeed, the critical threshold is determined bythe lowest lying well, an information usually unavailableto the investigator. Therefore, above a certain (but un-known) threshold of the injection energy, the measuredrelaxation times exhibit a marked nonlocal dependenceon the global potential profile. Such a nonlocal effect isdue to the contribution from slower, though rare, relax-ation trajectories, which explore the potential landscapesurrounding the well(s) of interest. Their presence can beappreciated, for instance, by looking at the distributionof the relevant relaxation times, though at the expenseof much longer observation times.The present analysis was restricted to 1D potentialsfor the sake of clarity, thus making our presentationhopefully easier to follow and affording higher numeri-cal statistics. Its extension to potentials in two and evenhigher dimensions confirms the overall picture summa-rized here and is presently matter of further investiga-tion. ACKNOWLEDGEMENTS We thank RIKEN’s RICC for computational resources.Y. Li is supported by the NSF China under grant No.11505128. P.K.G. is supported by SERB Start-up Re-search Grant (Young Scientist) No. YSS/2014/000853and the UGC-BSR Start-Up Grant No. F.30-92/2015 REFERENCES M. Goldstein, Viscous liquids and the glass transition: a poten-tial energy barrier picture , J. Chem. Phys. , 3728 (1969). H. Frauenfelder, S. G. Sligar, and P. G. 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