of absorption transitions (1, 2) and the mean rate of visits to state or are, respectively, J.=

Transcription

1 Proc. Natl. Acad. Sci. USA Vol. 85, pp , July 1988 Applied Mathematical Science Number of viit to a tate in a random walk, before aborption, and related topic (kinetic dim/multitate directional diagram/cy e/return proce) TERRELL L. HILL Laboratory of Molecular Biology, National Intitute of Diabete and Digetive and idney Dieae, National Intitute of Health, Betheda, MD 2892 Contributed by Terrell L. Hill, March 9, 1988 ABSTRACT Equation are derived for the probability of n viit to a given tate during the coure of a random walk on a finite diagram that tart from a pecified tate and end with aborption. By deriving the mean number of viit in two different way, certain conjecture or theorem are encountered that connect propertie of different but related dia in an intereting way. Other ubject included are (i) number of one-way tranition between two tate before aborption; (U) time dependence of the rate of cycle completion before aborption; and (ii) the relation of thi work to the "return proce" of arlin and Taylor. Thi i a continuation of two recent paper (1, 2). The ame notation and vocabulary are ued; it i aumed here that the reader i familiar with ref. 1 and 2. The principal problem conidered (in Section 1) i the number of viit to an arbitrary tate in a kinetic diagram (graph) before aborption occur at an aborption tate. Thi problem introduce ome intereting new mathematical propertie of diagram. Section 2 i a variation on Section 1. Section 3 and 4 are concerned with two pecial topic. Section 1. Number of Viit to a State Before Aborption of aborption tranition (1, 2) and the mean rate of viit to tate or are, repectively, J. a P, Joa Ea 'a, op Po, >i ax,,,. [1] The ubcript a refer to diagram A. The econd form for J.a arie from the teady-tate algebraic equation for tate o (except when or ). The mean time to aborption (1, 2) i then Fa 1/ and the mean number of viit to or before aborption i (oa # ) Joa na - 4 2j a(a) Here (ia) repreent the um of the directional diagram (3) for tate i in diagram A o that Pi (ia)/ia, where la i the um of directional diagram for all tate i in diagram A. Becaue of cancelation, 5a i not needed to calculate ha. Another way to expre ni i (o, #& ) O [2] Conider a continuou-time random walk on a finite diagram that tart (t ) at ome one tate and end at an aborption tate (there may be more than one tate). The tate i a non- tate. Let or be an arbitrary non- tate. What i the mean number H of viit to or, during the coure of the walk, before aborption occur? What i the probability frn of exactly n viit to cr? If a, the initial occupancy of a doe not count a a viit. The firt diagram, termed, in each of Fig. 1-3 provide an example. Thi i the "original" diagram (1, 2). Each tate ha a ingle precuror tate. The tate af may be entered from one or more neighboring tate deignated or'. A line between tate i and j in Fig. 1-3 repreent poible tranition i -.1 andj -+ i, with tationary tranition probabilitie at and aj,, repectively. An arrow indicate a oneway tranition with a imilar tranition probability. The tate probabilitie in the original diagram (1, 2) are p,{t) (ee Fig. 1). The original diagram i tudied, in effect, by enemble averaging. The diagram A (Fig. 1-3), called "cloed," can be ued to find teady-tate time-averaged propertie (1, 2) for the ame ytem. The tate probabilitie in diagram A are deignated Pi; thee are found from teady-tate algebraic equation (1-3). In the time-averaged repeated random walk, the mean rate The publication cot of thi article were defrayed in part by page charge payment. Thi article mut therefore be hereby marked "advertiement" in accordance with 18 U.S.C olely to indicate thi fact na tar pa/iv, That i, H. i the mean time pent in tate a per aborption divided by the mean time pent in ov per viit. Thu it i eay to find Ra from the cloed diagram A. However, A i not detailed enough to give nr,, (unle oa ): A mut be "ubdivided" into diagram B and C (Fig. 1-3) when ao. Define an "event" a either an aborption or a viit to o, The firt event tart at ; diagram B give timeaveraged propertie of repeated firt event only. The walk tart again at a oon a aborption or a viit occur. After the firt event, if aborption ha not occurred, the remainder of the walk tart from cr and end with aborption. Timeaveraged propertie from diagram C apply to thi part of the walk (all but the firt event). Diagram A i produced (from ) by dropping tate and running aborption arrow ( -- ) to tate. Diagram B i produced by dropping tate and o and running aborption and viit (ar'a-* ) arrow to tate. Diagram C i imilar to A except that aborption arrow run to tate ov intead of to tate. Note, in Fig. 1, where 3, 5, that not only i tate 4 miing from diagram B, but alo o,' S i miing. Thi i becaue, in thi cae, tate 5 i inacceible in the firt event. State probabilitie in diagram B and C are deignated Qj and Ri, repectively. Thee are found in the uual way (3). The "elementary" diagram, E (Fig. 1), i obtained by omitting all -I tranition from. There i of coure an E diagram for Fig. 2 and 3 a well, but thee are omitted. [3]

2 4578 Applied Mathematical Science: Hill Proc. NatL Acad Sd USA 85 (1988) oo of of x xi A a34 B C pa(t) Pi Qi \ 4 5- *6 C123 B 2/ 1)a A 3 5 aw 4 ~5 C E FIG. 1. Original diagram () and modification, in an example in which viit to a are counted. The rate of viit to a and of aborption in the repeated firt-event random walk (diagram B) are, repectively, E aan r' and >E aq d ~~~~~~~ Hence, the probability r that the firt event end in a viit to ar i E aff(o'b) r [4] > a, (o'b) + E O(B)4 Here, a in Eq. 2, lb ha been canceled from Qj (ib)/ab. Correponding to Eq. 1, we have, for diagram C, J >E ar, c I aorara Jc + c Ra I a,,. [6], Eq. 6 i the teady-tate equation for tate oa. We then have for ir, zo 1 - r \ n-1 n + + (n -1). [7] Uing Eq. 5 and 6, thi probability ditribution lead to bc r(jc + c i c r( a4,,(rc) ( E: a( C) I The ubcript bc i a reminder that thi 7 i found from diagram B and diagram C. If or, diagram A i the only diagram and it uffice to give it,,: r ( j( djra ) ( a )" (n 2 ). [9] Alo, Eq. 2 need modification in thi cae: Ea av,'ao-'a) na -a (A E a(a)' [8] [1] am_ B' FIG. 2. Original diagram () and modification, in an example in which viit to a, are counted. The equality nf hbc (Eq. 2 and 8) ha ome intereting conequence. In the firt place, if there i only one aborption tate (a in Fig. 1 and 2), the two denominator in Eq. 2 and 8 are equal becaue a cancel and (A) (C). The latter equality follow becaue diagram A and C are the ame except for the final detination of the one-way arrow (a) out of tate. But a one-way arrow out of make no contribution to the directional diagram of tate, which are contructed from arrow leading into. Hence (A) and (C) are both the ame a (E); recall that diagram E (Fig. 1) ha no one-way arrow. Uing (A) (C), Eq. 2 and 8 lead to r (oa)/(aqc). Thi expree r in term of the propertie of A and C; on the other hand, Eq. 4 relate r to diagram B. Thi point will be returned to below. Actually, a number of pecial cae with two or three aborption tate () have been examined in detail and it wa found in each cae that a56 E a(a) >E a( C). [11] Thee are the denominator in Eq. 2 and 8. The um were found to be equal but not, in general, the individual term for each. Eq. 11 i a conjecture at thi point; a general theorem i proved in the appendix. I will proceed on the aumption that Eq. 11 i correct and make further comment on thi ubject at the end of thi ection. CR' a B > a15 4 al5 A 4 3 ~4 3 4 a48 FIG. 3. Original diagram () and modification, in an example in which viit to oa are counted.

3 Applied Mathematical Science: Hill If Eq. 11 i accepted, then r (o-a)/(ac) [12] i a general reult (not retricted to a ingle tate). Again, after an examination of pecial cae, the relationhip of Eq. 12 to Eq. 4 appear to be (oa) > aa,(c'b) [13] (crc) a,,,-(o-'b) + E a(b) [14] (arc(o~a) + >1 a([15 B). O-C) (ora) + Ea ( B). [15] Thee are conjecture; general proof have yet to be etablihed. Obviouly, if Eq. 13 and 14 can be proved, then Eq. 11 follow from Eq. 2 and 8. However, Eq. 11 i eaier to prove than Eq. 13 and 14. Eq are intereting connection among diagram A, B, and C. Note that B i not ymmetrical with repect to A and C (ee the dicuion of B' below). It hould be mentioned that, for a cae uch a Fig. 1 in which one or more o tate drop out in diagram B, Eq. 13 and 14 no longer hold eparately a written. Rather, the ratio (oaa)/(crc) i equal to the ratio of the right-hand ide of Eq. 13 and 14. There are further relation among diagram A, B, and C. We have already introduced Ta 1/J,. Similarly, Tc 1/Jr i the mean time to aborption after the firt event, if the firt event i a viit to oa. The mean duration of the firt event itelf i Tb aooq + E aq] 1b/(C), [16] having ued Eq. 14. The mean time to aborption (Ta) ha two contribution, and the ame i true of the fraction of time pent in tate i: Ta Tb + rtc [17] Pi (Tb/Ta)Qi + (rtc/ta)ri. [18] Becaue tate a i miing from diagram B, Q,. Another imple relation, which follow from Eq. 1, 5, and 11, i tc/ ta 1;c/1-a. [19] On ubtituting Pi (ia)/ia, etc., into Eq. 18, one find for an arbitrary non- tate i, (o-c)(ia) (o-a)(ic) + (ib) I a.(a). [2] Summation of Eq. 2 over the tate recover Eq. 15. Summation over all i give (o-c)i. (o-a)lr + Yb E a( A). [21] If the role of and cr are exchanged in (e.g., in Fig. 2), that i, the walk tart at cr and viit to are counted, then diagram A and C are exchanged and diagram B become diagram B' (Fig. 2). The ue of i in Eq. 2, followed by the, ar exchange a jut decribed, lead to the intereting reult (B) (crb'). In Fig. 2, (1B) (3B'). Thi i eaily Proc. NatL. Acadc Sci USA 85 (1988) 4579 verified. In imilar fahion, from Eq. 21, one find the ymmetrical relation [(A)Y-b + (TA)Mb']z;C [(SC)Yb + (ctcab'tha. [22] Finally, in thi ection, I return to dicu Eq. 11 further. Given a diagram, with it -- tranition or tranition, diagram A and C follow on aigning and acr(o- ). In fact, Eq. 11 relate to any aignment of and cr. Thu, we can be more general. Define diagram Au a that obtained from by omitting all tate and running all -- one-way arrow to tate u, where u i any non- tate. The theorem to be proved (ee the Appendix), then, i that the um I a( A") [23] i the ame for every non- tate u in. If there i only one tate, the theorem obviouly hold: the ingle term in the um i equal to a(e), which i unrelated to the choice of u. The generalization of Eq. 19 i that t4 i proportional to X"' for any u, with the ame proportionality contant. A an aid in proving the invariance of 23, I have noticed the following (in a number of example). Firt, define P-tate directional diagram a a generalization of the uual (onetate) directional diagram: arrow flow toward one or more of v deignated tate. Then, for example, if 1, 2, or 3 in a diagram E (derived from ), we contruct from E the full et of one-tate ( 1, 2, or 3), two-tate ( 1, 2; 1, 3; or 2, 3), and three-tate ( 1, 2, 3) directional diagram. Thi full et can then be ued, term by term (by inerting a factor), to generate the term in 23, which indeed are algebraically equal for any u. Section 2. Number of One-Way Tranition Between Two State Before Aborption A a variation on Section 1, one can ak how many tranition i -- j occur between two tate before aborption occur. The dicuion i confined to a ingle example. In the diagram in Fig. 4, how many (n) tranition 3 2 occur before aborption , if the walk tart at 1? The procedure in Section I i changed very little. We can find fia eaily from the cloed diagram A, but to obtain the probability ditribution ir,, we again need the ubdiviion of the timeaveraged walk into diagram B (firt event) and C (ubequent to firt event). An event here i either a tranition 4 -* 5 (aborption) or a tranition 3 -* 2. In diagram B, either tranition return the walk to 1. If the firt event i 3 -* 2, then the remainder of the walk tart from tate 2 (diagram C) İn diagram A, a45p4, Jt a32p3, oji 2 3 a45 1 ~4 B ~32 Q; FIG o- 3 a23 A Jta _ a32(3a) a45(4a) [24] a45 1 ~4 Pi Example in which tranition are counted.

4 458 Applied Mathematical Science: Hill where t refer to the tranition There are five term in (3A) and four term in (4A). In diagram B, the probability that the firt event i 3 -> 2 i r - a32q3 a32(3b) [25] a32q3 + a45q4 a32(3b) + a45(4b) There are five term in (4B), and (3B) (3A). In C, the two fluxe are Jc a45r4 and Jtc a32r3. [26] Alo, (3C) ha ix term, and (4C) (4A). The expreion for 7T, i then Eq. 7 with t (ubcript) in place of o. Then, a in Eq. 8, nb r(jc + Jtc) r[ a45(4c) + a32(3c)] a45(4c). Examination how that a32(3b) + a45(4b) a45(4c) + a32(3c), [28] and hence that Ha bc a expected. Yi-der Chen (peronal communication) ha encountered imilar problem in hi work on exchange cycle completion and tracer fluxe. Section 3. Time Dependence of Rate of Cycle Completion Here I expand briefly on a comment made in the dicuion of figure 6a of ref. 2. Conider an original diagram with cycle and with aborption. nowledge of the time dependence of the tate probabilitie in thi diagram i not ufficient to provide the time dependence of the rate of cycle completion of the variou type q (e.g., q a+, a-,....). For thi purpoe, the cloed-2 diagram (1, 2) mut be formed. In thi more detailed diagram the individual one-way cycle are expoed and the aborption tranition are till preent (in fact, generally multiplied). The time dependence of the tate probabilitie in thi diagram are now ufficient for the purpoe at hand. For example, uppoe the one-way tranition i -*j correpond to completion of a cycle of type q. Then the rate of completion of cycle of thi type at t i ajp,(t). The integral of thi quantity from t to oo give the mean number of 'q cycle completion before aborption. A a check, form the cloed diagram (1, 2) from the cloed- 2 diagram (thi reult in a cloed-3 diagram, a in figure 9c of ref. 2). Let Pi be the teady-tate probability of tate i in the cloed-3 diagram. Then the mean number of q cycle completion before aborption i (2) aupif au f pi(t)dt. [29] It hould be noted that the equence of diagram manipulation here i the revere of that ued in figure 9 of ref. 2. Section 4. The "Return Proce" of arlin and Taylor For original diagram with aborption at tate, I have been uing (1, 2) the auxiliary cloed diagram ( -* tranition go to the tarting tate and tate are eliminated) to obtain time-averaged mean propertie of the original random walk without having to contend with the time dependence of enemble average in the original diagram. For a dicretetime proce, arlin (p. 89 of ref. 4) and arlin and Taylor (p. 112 of ref. 5) alo ue an auxiliary return to but in a different way that doe not have the ame phyical ignifi- cance. In particular, they retain tate in the auxiliary proce and introduce an extra tranition -> with probability unity. In a long random walk on their. auxiliary diagram (tationary probabilitie Pi), after each aborption at a tate, there i a one-tep ( --+) delay until the walk tart over again at. If J i the rate of tarting walk at, [3] where 7 i the mean (dicrete) time to aborption. It i eay to devie a continuou-time verion of the above. For the auxiliary diagram, introduce a one-way tranition -+ for each tate, each with tranition probability x (arbitrary). The tate remain in the auxiliary diagram. Let Pi be the tationary probabilitie of tate. In a long random walk on the auxiliary diagram, the rate of new tart from i J NXP. The mean time per tart i F (mean time to aborption at a tate, tarting from ) plu 1/x, the latter being the mean delay time introduced by the extra tranition -* S. Thu, or Proc. NatL Acad ScL USA 85 (1988) i I P. F + 1 i/j9 J J xz P, 1 P x XIE P ~1 _1 t +- x x J [31] [32] Of coure t, depite appearance, i independent of x. In effect, in our cloed diagram (1, 2), x -* oo. APPENDIX A formal proof of 23 i given here. Thi proof wa devied by Doron Zeilberger (Drexel Univerity, Philadelphia), baed on the uggetion made following 23. We conider directed graph, poibly with multiple edge and loop. A panning tree of G (diagram) rooted at a vertex u i a ubgraph of G that ha all it vertice and ome of it edge and i a rooted tree uch that all arrow go toward u. Note that in any rooted tree there i a unique path from any vertex to the root. Define the weight of an edge u -- v to be the indeterminate a., and the weight of a panning tree to be the product of the weight of all it edge. 23 can be retated a follow. THEOREM: Let G be a graph with I aborbing tate j (i 1,..., I), uch that each ofthe j ha exactly one predeceor j. Let A a,!. For any nonaborbing vertex u ofg, let Gu be the reduced graph obtained by deleting all the aborbing vertice jand redirecting the edge ; -p jto become the edge ;- u, with the ame weight; i.e., weight (l- u) Let. (,!G.) be the um ofall the weight ofall the panning tree of Gu, rooted at ;. Then i i1 [33] i independent of u. Proof: Let G be the graph obtained from G by deleting all the aborbing vertice i (i 1,..., I), adding a new vertex t, and replacing all the edge ;- i by,- t, with the ame weight: weight (i - t) /3i. Let um be the um of all the weight of all panning tree

5 Applied Mathematical Science: Hill of G, rooted at t. It will now be hown that, for every u in G, E AN,4G,) um. [34] Thi will prove the theorem ince um i independent of u. A one-one correpondence between the term of the left ide and the term of the right ide of Eq. 34 will be preented. A typical term on the left i 8i -weight(t), for ome panning tree T of GU, rooted at ;'. To thi tree we aociate a panning tree T, of G, rooted at t, a follow. Add the edge, t, and replace all the edge of the form j- u by edge -! - t. It i eaily een that f3i weight(t) i equal to weight(t) ince by definition the weight of j-- u and j-+ t are the ame, namely 1p. The new edge ;- t i accounted for by the factor (,i in Si -weight(t). It i eaily een that T i alway a bona fide panning tree of G rooted at t. Note that can be recaptured from Tby oberving that it i the predeceor of t in the unique path from u to the root t. Thi give u the revere correpondence. Given a pan- Proc. NatL Acad Sci USA 85 (1988) 4581 ning tree t, rooted at t, of G, aociate to weight(t) the following term on the left. Let be the predeceor of t in the unique path connecting u and t, and let T be the panning tree of Go obtained by deleting t and replacing each edge of the form!- t (j t) by the edge j-- u. We aociate the term weight(t) on the right with the term,bweight(t), and it i obviou that they are identical. We have thu found a one-one correpondence between the term of the left ide and the term of the right ide, and thi etablihe Eq. 34, and thu prove the theorem. I am indebted to Dr. Yi-der Chen, Howard Taylor, and Doron Zeilberger for helpful comment in the coure of thi work. 1. Hill, T. L. (1988) Proc. Natl. Acad. Sci. USA 85, Hill, T. L. (1988) Proc. Natl. Acad. Sci. USA 85, Hill, T. L. (1977) Free Energy Tranduction in Biology (Academic, New York). 4. arlin, S. (1969) A Firt Coure in Stochatic Procee (Academic, New York). 5. arlin, S. & Taylor, H. M. (1975) A Firt Coure in Stochatic Procee (Academic, New York), 2nd Ed.