Universality of Poisson Indicator and Fano Factor of Transport Event Statistics in Ion Channels and Enzyme Kinetics

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1 pub.ac.org/jpcb Univerality of Poion Indicator and Fano Factor of Tranport Event Statitic in Ion Channel and Enzyme Kinetic Srabanti Chaudhury,*,, Jianhu Cao, and ikolai A. Sinityn, Theoretical Diviion, Lo Alamo ational Laboratory, Lo Alamo, ew Mexico, United State ew Mexico Conortium, Lo Alamo, ew Mexico, United State Department of Chemitry, Maachuett Intitute of Technology, Cambridge, Maachuett, 39 United State ABSTRACT: We conider a generic tochatic model of ion tranport through a ingle channel with arbitrary internal tructure and kinetic rate of tranition between internal tate. Thi model i alo applicable to decribe kinetic of a cla of enzyme in which turnover event correpond to converion of ubtrate into product by a ingle enzyme molecule. We how that meaurement of tatitic of ingle molecule tranition time through the channel contain only retricted information about internal tructure of the channel. In particular, the mot acceible flux fluctuation characteritic, uch a the Poion indicator (P) and the Fano factor (F) a function of olute concentration, depend only on three parameter in addition to the parameter of the Michaeli Menten curve that characterize average current through the channel. everthele, meaurement of Poion indicator or Fano factor for uch renewal procee can dicriminate reaction with multiple intermediate tep a well a provide valuable information about the internal kinetic rate. I. ITRODUCTIO Quantitative biology i aimed at developing mathematical/ theoretical tool for quantitative prediction of biochemical ytem dynamic. Thi field ha alway been influenced by the problem of a large diverity of biochemical procee. Even if one develop a very precie decription of ome kinetic biochemical procee in one organim, it i uually unlikely to find exactly the ame biochemical proce in another organim. Hence, the unifying law that encounter in wide range of biochemical ytem are of particular importance for thi field. One widely known example i called the Michaeli Menten (MM) law. According to it, the average rate of product creation J and the ubtrate concentration [S], in enzymatic reaction, are related by k[s] J = [S] + KMM () where K MM and k are contant parameter. The MM law wa initially derived for a imple reaction k k E + S Hooo I ES P + E k E in which the ubtrate S i converted into product P via an intermediate complex ES that the ubtrate create with enzyme molecule E. Interetingly, the MM law wa found in a much wider cla of enzymatic reaction, with poibly many internal ubtep. Recently, thi obervation wa explained by howing that a large cla of paive (i.e., driven only by difference of ubtrate and product concentration) enzymatic reaction, with multiple tate of enzyme ubtrate complex, follow the Michaeli Menten law. 4,5 Equation i alo encountered in biochemitry beyond the context of enzyme kinetic. For example, traditionally, the key meaurable quantity in ion channel tranport ha been the teady tate flux in a ingle channel through a membrane that eparate two compartment with different olute concentration.,3 MM law wa found to decribe the tranport of olute molecule through a cla of ion channel, in which J repreent the ion flux through the channel and [S] i the olute concentration on one ide of the channel, auming that [S] = on the other ide. Recent theoretical tudie hed light on the origin of oberved wide applicability of the MM-formula (eq ) in ion channel. Bezrukov et al. 3 howed that a general model of tranport through a chain of neighboring ite with ( ) rate contant, a well a it continuou D diffuion limit, produce the ame dependence of average flux on olute concentration a in an effective MM model with appropriately choen tranition rate contant. To avoid a mixture of ion channel and enzyme terminology, in thi article, we will ue the ion channel interpretation of our model throughout the text and return to enzyme application only in the dicuion. Received: September 8, Revied: ovember 7, Publihed: December, American Chemical Society 53 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59

2 The implicity of eq implie that information provided by meaurement of average flux i intrinically limited. Only two contant can be obtained experimentally by meauring MM curve for a renewal proce. Additional term will contribute to the ubtrate-dependence of turnover rate if a multiple-channel reaction i driven away from equilibrium. 4,5 The advance of ingle molecule technique allowed reearcher to alleviate thi retriction by tudying not only average current but alo the tatitic of ingle molecule tranition in the channel-facilitated tranport through biological membrane.,3,6 8 For example, ingle-channel ion current meaurement have been ued to tudy the tranlocation dynamic of ingle-tranded RA and DA through α-hemolyin channel in lipid bilayer membrane. 9 During the tranlocation, the ingle tranded polymer partially block the channel. Thi lead to tranient blockade in α- hemolyin ingle channel current and the current i retored to it original value when the DA exit from the other ide of the membrane. By detecting time moment of uch event of DA exit, one can tudy not only mean DA tranition time but alo characterize fluctuation of thoe time interval. Such highreolution tranition event recording from ingle ion channel were hown to provide information that i hidden in enembleaveraged experiment. The mot acceible characteritic of fluctuation in molecular tranport are related to econd moment of turnover time tatitic and current ditribution. Thoe include the Poion indicator (P), defined by = t P t t which i alo known a the Mandel parameter in the context of photon counting tatitic, and the Fano factor (F), defined by = J J F J where t i the time between ucceive molecular tranition through the channel and averaging i over a large number of uch oberved tranition; J i the total number of molecule tranferred through the ion channel during a pecified meaurement time interval. In the context of photon tatitic, the Fano parameter and the Poion indicator are related to the Mandel parameter which decribe the bunching and antibunching of emitted photon. If ion channel were ut window without internal tructure for intantaneou molecular tranition through a membrane, the tatitic of turnover time would be exponential and tatitic of current would be Poionian. Thi correpond repectively to P = and F =. Functional dependence of P([S]) and F([S]) on the olute concentration, [S], on one ide of a membrane provide valuable information about tructure of an ion channel. In our preent paper we will conider the model of ion channel kinetic with the poibility of multiple cloed loop in kinetic network, a hown in Figure. We will how that, imilarly to the univerality of average flux characteritic, the complexity of P and F for tranport through ion channel reduce to the univeral function that depend on, maximum, three additional contant parameter. We will alo how that one can derive a connection between the Fano factor and Poion indicator for uch renewal kinetic procee. Although the univerality of P and F will be the main focu of our work, we will perform calculation on the level of the full () (3) Figure. Ion channel model. E i the empty tate with internal tate for the entry to the ion channel. tatitic of turnover event, o that our method can be ued to explore imilar propertie of higher order correlator, if needed. Thi paper i organized a follow. In ection II, we introduce our general kinetic model for tranport through an ion channel and derive expreion for the firt paage time ditribution and related obervable in fluctuation tatitic uch a the econd moment of the ditribution and the Poion indicator. We alo concentrate on imple kinetic model with only two internal tate and ue the elf-conitent pathway formalim propoed by Cao and Silbey 5 to derive expreion for the waiting time ditribution in term of the elementary kinetic rate and how how thee model influence parameter in the general expreion for the Poion indicator. In ection III, we calculate the Fano factor by exploring a connection between the turnover probability ditribution and the cumulant generating function, which i directly related to the Fano factor. We derive expreion for the parameter that influence Fano factor in term of the kinetic rate for two tate model. We ummarize our reult in ection IV. II. FIRST PASSAGE TIME DISTRIBUTIO In our model of ion channel, hown in Figure, we conider tranport through a ingly occupied channel with arbitrary number of internal tate. The channel i aumed to be capable of having a maximum of one molecule inide; that i, even if the molecule i maller than the length of the ion channel, we aume that it create a potential that block other molecule from entering the channel. The left ide compartment contain olute particle at concentration [S], wherea the right compartment ha negligible olute concentration. E repreent the ingle empty channel tate, and P, P,..., P are poible olute occupied tate at the entrance to the channel. We aume that, after the olute molecule leave the channel, the internal degree of freedom of the channel relax quickly, o that the empty tate E of the channel can be repreented in our model by a ingle tate. We alo aume that the olute concentration on the left of the channel i et to a contant value. Hence, evolution repeat, in the tatitical ene, each time the channel become empty. If we undertand the dynamic between only two ucceive moment at which the channel become empty, we can recontruct all other tatitical characteritic of the proce. Thi type of reaction cheme i referred to a a renewal proce. 6 Waiting Time Ditribution Function. We aume that experimentally only ome pecific event, called monitored tranition, are obervable. For example, in ion channel, monitored tranition can be event when a tranported 54 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59

3 molecule leave the ion channel. We will aume that the evolution of the ytem are tatitically identical after each monitored event. The central obect of the renewal theory i the firt paage time ditribution ϕ(t) between two ucceive monitored event. More preciely, given the moment of one monitored tranition, ϕ(t) dt i the probability of oberving the next monitored tranition between time t and t +dt after thi time moment. In thi article, we will aume that monitored tranition correpond to event when olute molecule are leaving the ion channel to the right compartment. Such event were hown to be detectable in ion channel experiment.,3 It i generally aumed that the kinetic rate for entering the empty ion channel i proportional to the olute concentration [S]. Hence let k E [S] be the rate for making a tranition from empty tate E into the tate with a olute molecule inide the ion channel at ite, with contant parameter k E independent of [S]. In correpondence to thi proce, we introduce the probability per unit time, Q E (t) of the event that at time t, after the channel become empty, the new olute molecule enter the channel for the firt time at ite. Explicitly, Q E (t) i exponentially ditributed Q () t = k [S]exp( t k [ S]) E E E = (4) Thi type of probability i known a the waiting time ditribution function, which account not only for fundamental rate procee or their combination but alo nonexponential decay procee. The waiting time ditribution formulation allow u to condene a large cla of complex reaction into a generic cheme which i the irreducible repreentation of meaurement. 5,7 In the current etting, the acceible meaurement are ubtrate binding and enzymtic turnover, which define the baic element of the waiting time analyi preented below. Let then Q R (t) be the probability per unit time of the event that the molecule that ut entered the ite will leave the channel at time t to the right, i.e., by making a monitored tranition. We aume that all elementary reaction, except the monitored one, are, in principle, reverible, o there i alo a finite probability per unit time, Q L (t), that the olute molecule, being initially at ite will leave the channel to the left at time t without making the monitored tranition. In both cae, after leaving to the left or to the right, the channel become open again and the proce renew. The probability ϕ(t) then atifie a formal convolution law ϕ() t = d t Q ( t )[ Q ( t t ) = t t E R + d t Q ( t ) ϕ( t t )] L t (5) Thi equation can be tranformed into algebraic equation, which i atified by the Laplace tranform of ϕ(t), i.e., by ϕ() = e t ϕ(t) dt. The reult i ϕ() = Q ()[ Q () ϕ() + Q ()] = E L Here and in what follow, we ditinguih probability ditribution and their Laplace tranform by writing in their argument, repectively, t or. Equation 6 can be formally olved a R (6) ϕ () = = E R Q () Q () Q ( ) Q ( ) = E L (7) Thi compact expreion generalize the ditribution function derived for chain reaction and exemplifie the elf-conitent pathway method formulated for the firt paage time ditribution of generic enzymatic network. 5,7 The introduction of Q R and Q L implifie uch analyi and can generate hierarchical ditribution function for chain reaction. Subtrate Dependence of Poion Indicator. We note that eq 7 i till a formal olution becaue only the functional form of Q E (), at thi tage, i known ke[s] Q () = E + = k [S] E (8) while Q L () and Q R () remain unknown yet. However, to achieve our goal, their explicit form i not needed. Importantly, we know that neither Q L () nor Q R () depend on the external olute concentration [S]. Subtituting Q E () from eq 8 into eq 7 and taking the derivative of eq 7, t =( )lim ϕ()/, we obtain the average firt paage time A t = + B [S] (9) where A = / ka B = ( b + c)/ a () where a = Q R (), a = Q L (), b and c are contant that are the firt derivative of repectively Q R () and Q L () at =. Thu, we obtained a linear relation between the mean firt paage time t and the invere of olute concentration [S]. Thi i equivalent to the relation in eq obtained for J =/ t. For example, parameter of the MM curve, K MM and k, can be expreed via A and B a K MM =A/B and k =/B. Awe mentioned, thi univerality, i.e., independence of functional form of J ([S]) on the detail of the internal kinetic model of the channel, wa previouly dicued in a erie of previou work. 3,5 ext, by analogy with average turnover rate we conider higher moment of the turnover time ditribution. Subtituting eq 7 and 8 into t n = dtt n ϕ(t) =( ) n lim n ϕ()/ n,wefind that the Poion indicator P defined in eq read [S]( q[s] η) P = ( δ + [S]) () where q, η, and δ are all contant that are different combination of the different rate contant k E and the firt and econd derivative of Q L () and Q R () at =, which do not depend on olute concentration [S]. Explicitly, we obtained A = / ka, B = ( b + c)/ a, q = b bc + a( d + f)/( b + c), η = / b k ( b + c), δ = / k ( b + c) d and f are the econd derivative of repectively Q R () and Q L () at =. 55 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59

4 Equation, a well a the imilar expreion for the Fano factor that we will derive in the following ection, are the central reult of our work. Equation how that the Poion indicator P ha a univeral functional dependence on the olute concentration. It i parametrized only by three contant parameter irrepective of the number of internal tate and kinetic rate inide the ion channel. The parameter P i, generally, a nonmonotonou function of [S], and at high olute concentration, lim [] P = q; that i, generally at high concentration the tatitic of turnover time ditribution i non-poion. ote that eq i derived under the aumption that ubtrate binding a decribed by eq 8 i a rate tep. A general functional form of [S]-dependence can be obtained by incorporating the non-poionian ditribution of the ubtrate binding. C. Kinetic with Two Internal State. Recently Cao and Silbey 5,7 propoed a elf-conitent approach, which i baed on the theory of renewal procee, for tudie of turnover time tatitic in ingle molecule kinetic. Thi theory i equally well uited for application to ion channel. It provide a traightforward way to expre waiting time ditribution Q(t) via the elementary kinetic rate of a kinetic model. Although generally explicit expreion would be complex, uch expreion can be eaily written for implet model with only one and two internal tate. A an example, conider the model with only two internal tate, hown in Figure a. E repreent the empty tate, and ES Figure. Two-tate ion channel model: (a) E i the empty tate, and ES and EP are the two internal tate. The forward and backward rate contant for tranition between E and ES are, repectively, k and k E ; k + and k are rate contant for intrachannel tranition between ES and EP, and k i the ecape rate from the channel. (b) ES and ES are the internal tate. The ecape through the channel take place from ES. and EP are the two interconvertible internal tate, which correpond, e.g., to two internal tate of a molecule inide an ion channel. By applying the theory, 5,7 the elf-conitent equation for the firt paage time ditribution in the Laplace pace i Figure 3. Poion Indicator P a a function of the olute concentration [S] for the two-tate ion channel model in Figure a. umerical parameter value are k =,k E =,k + =., k =., k =, and q =.6 (blue); k =,k E =,k + =., k =., k =, and q =.89 (green). For ion channel model in Figure b: k =,k E =, k + =,k =.7, k =.5, and q =.35 (black), and ion channel model with one internal tate (MM model): k =,k E =,k =, and q = (red). Q () Q () Q () 3 ϕ () = Q ( ) Q ( ) Q ( ) Q ( ) E () where Q 3 () decribe the monitored tranition, Q () and Q () decribe the tranition from the tate E to ES and the tate ES to EP, repectively. Q E () and Q () are the backward tranition rate from the intermediate tate ES to the empty tate E and the tranition from the tate EP to ES, repectively. In analogy to eq 7, Q 3 () and Q () repreent Q R () and the backward tranition Q 3 () and Q () repreent Q L (). Equation for thi two tate kinetic i thu a pecial cae for eq 7. Following eq 8, the waiting time ditribution in term of the kinetic rate contant are given by Q () = k [S]/( + k [S]) Q = k /( + k + k ) E E E Q () = k /( + k + k ) + E + Q () = k /( + k + k ) Q () = k /( + k + k ) + 3 (3) Taking the derivative of eq and uing eq 3, we found that parameter q, η, and δ in eq in term of kinetic rate of Figure a can be expreed a kk + q = ( k + k + k ) η = δ = + kk + ( ke + k + k+ + k ) k ( k + k + k ) + kk + + ke( k + k ) k( k + k+ + k ) (4) 56 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59

5 The negative value of the parameter q mean that fluctuation of turnover time for thi two tate model are alway ub- Poion; that i, they are uppreed in comparion to the one in the Poion proce. In contrat, for the reaction proce hown in Figure b, in which the econd internal tate i an idle tate, uing the elf-conitent approach a before, we find that q =k + k /(k + + k ). Thi poitive value correpond to uper- Poionian tatitic. We note that kinetic model with more than two internal tate may till be ditinguihable from two-tate model if variance of fluxe are meaured. For example, for the reaction cheme given in Figure a, the minimum of the Poion indicator i achieved at P min η = 4( δη+ δq) (5) Minimizing thi expreion further over the choice of kinetic rate, we find the minimum poible value of the Poion indicator i /3, and it i achieved for reaction in Figure a when all elementary reaction are irreverible and have the ame kinetic rate k k k E ES EP P Hence the value of P min lower than /3, if oberved, would indicate that the reaction mechanim involve more than two intermediate tate. III. CURRET DISTRIBUTIO FUCTIO AD FAO FACTOR A different type of ingle molecule meaurement i the probability ditribution for the number of event oberved within a time bin. 8,9 In thi meaurement approach, the number of molecule tranferred through the channel i meaured during time interval t and the probability P n (t) of the number n of tranition i obtained after many repetition of the meaurement. A convenient way to tudy current ditribution theoretically i by introducing the probability generating function (pgf) in χ Z( χ, t) = P( t)e = e n= n ω( χ) (6) where χ i called the counting parameter and ω(χ) i the cumulant generating function. It derivative with repect to χ give the cumulant of the ditribution P n, uch a the mean n and the variance σ = ( ) n i σ, = = ( ) n n ( i) (7) The Fano factor i defined to be the ratio of the variance to the mean, i.e. F = ( i) ( ) i ( ) (8) Recently, the Fano factor in mot general enzymatic model with two internal tate, which alo correpond to our model of ion channel with two internal nonempty tate, wa tudied by Mugler et al. 3 It wa hown that meauring both the average current and the Fano factor a a function of olute concentration i ufficient to ditinguih among all poible two-tate enzymatic kinetic model and, moreover, to determine value of all kinetic rate quantitatively. The natural next quetion i whether meaurement of the Fano factor can be ued to extract information about more complex enzymatic mechanim. To reolve thi quetion, we will firt demontrate the connection between the turnover probability function and the cumulant generating function, from which the Fano factor can be readily obtained for renewal procee. Let ϕ(t) be the probability that a turnover event take place in time t and ψ(t) i the probability that no monitored tranition happen during time t after the lat monitored event. Then the event averaged probability ditribution function P n (t), after the Laplace tranform over time, i given by Pn( ) = ϕ () ψ() n (9) Uing eq 9, the generating function in the Laplace pace become n iχn ψ() Z( χ, ) = ϕ ( ) ψ( )e = iχ e ϕ( ) n () = which i the dicrete Fourier Tranform of P n () over n-index and it Laplace tranform over time. Returning to real time i t Z( χ, t) = e i ψ() i e χ ϕ( ) () At large meaurement time, we look for the dominating exponential part in eq. Thi happen at = * where * i the pole in the denominator in eq which i provided by the olution of the equation iχ e ϕ( * ) = () A a function of counting parameter, the olution of eq alo coincide with the cumulant generating function, defined in eq 6 beaue, at large meaurement time t, according to eq and, the generating function behave a Z(χ,t) e *(χ)t. Thi form correpond to linearly growing current cumulant and, hence, a contant value of the Fano factor. For illutration, in Appendix A, we perform calculation of all function for a imple MM model explicitly uing thi approach. The relation between the turnover time probability ditribution ϕ(t) and the cumulant generating function 5 ugget that one can expre the Fano factor in term of derivative of ϕ(t) and obtain a imilar univerality to the Poion indicator. Indeed, from eq we have χ ϕ () = e i (3) The invere of thi function can be written a χ i = ϕ (e ) = ω( χ) (4) Uing the propertie of invere function and it derivative a hown in Appendix B, the Fano factor F i given by 57 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59

6 F = ϕ() ϕ() = ϕ() (5) Applying eq 6 to eq 5 we obtain α β γ = + + [S] [S] F ( μ + [S]) (6) Contant parameter in eq 6 are not independent. Additional contraint on them follow from the fact, which wa etablihed in the previou ection, that the flux mut be Poion ditributed (F ) when [S]. Thi lead to αb[s]([s] KB) F = ( μ + [S]) (7) where ( bb+ c) ad ( + f) αb = ( b + c) / b ke KB = ( bb+ c) ad ( + f) μ = ke ( b + c) (8) Parameter a, b, c, d, and f in eq 8 were introduced in the previou ection. Conidering eq 7, we conclude that, imilarly to the Poion indicator, the Fano factor i parametrized only by three independent contant. Equation 7 how that the Fano factor F ha the ame functional dependence on the olute concentration irrepective of the number of internal tate inide the ion channel. For the two tate model in Figure a, parameter α B, K B, and μ can be explicitly written in term of elementary kinetic rate kk + αb = ( k + k + k ) + ke + k + k+ + k KB = k kk + + ke( k + k ) μ = k( k + k+ + k ) (9) Figure 4 how the dependence of F on olute concentration [S] for the reaction cheme hown in Figure a,b and for the ion channel model with only one internal tate (MM reaction). A in the cae with Poion indicator, competing reaction cheme can be ditinguihed baed on the value of the Fano factor. For example, the value of F lower than /3 i an indication of a reaction cheme involving more than two intermediate tep. IV. COCLUSIO In thi work, we howed that the Poion indicator and the Fano factor have imple generic functional dependence on olute concentration irrepective of the number of internal tate in the ion channel kinetic model. Thi obervation can be ued in practice by analogy with application of the MMformula. For example, many biochemical procee favor Figure 4. Fano factor F againt olute concentration [S]. Parameter for Figure a are k =,k E =,k + =., k =., and k = (blue). For Figure b: k =,k E =,k + =,k =.7, and k =.5 (green) and MM reaction (red): k =,k E =,k =. enzyme with pecific value of contant K MM and k. 3,4 In addition, by looking at k/k MM value, one can compare enzyme preference for different ubtrate. We anticipate that meaurement of parameter of P([S]) and F([S]) curve can have imilar application. oie ha been hown to lead to important conequence in biological ytem. While ome biological procee need to uppre noie, other may need noie a an important part of the reaction mechanim. 5 9 It hould be intereting to explore parameter that characterize the Poion indicator and the Fano factor curve in a wide cla of ion channel and enzymatic reaction. One can expect that evolutionary election ha led to eparation of enzyme and ion channel in clae with parameter that either uppre or enhance noie for pecific biological reaon. The univerality of flux fluctuation impoe retriction on the information about the tructure of a tudied ion channel that can be obtained by meauring variance of tranport characteritic. On the other hand, P([S]) and F([S]) curve allow u to ditinguih reaction cheme and extract ome combination of kinetic rate from experimental data. In thi work we limited our dicuion to renewal procee in which the empty tate of the channel i repreented by a ingle tate. Extenion of our formalim to other kinetic cheme, uch a nonrenewal procee with multiple interconvertible empty tate, can be a ubect of the future reearch. APPEDIX A. EXAMPLE: MM KIETICS Let u conider a MM enzyme catalytic reaction k k E + S Hooo I ES P + E k E For thi imple reaction, following the elf conitent approach, the waiting time ditribution can be written a 58 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59

7 k[s] Q = + k[s] k Q = R + k + ke ke Q = L + k + ke (A) Subtituting eq A in eq 6 we obtain kk [S] ϕ () = k + ( k + )( k [S] + ) and the Poion indicator i given by k[s] P = k+ ke k + [S] E (A) ( k ) Subtituting eq A in eq, we find (A3) *= k + k + k ( ( [S] E ) iχ + ( k[s] + ke + k) + 4 k[s] k(e )) (A4) Thi reult coincide with the cgf obtained previouly by olving the mater equation for the generating function. APPEDIX B Let y = ϕ() =e iχ, then = ϕ (y) =ω(χ) The firt and econd derivative of the invere function are given by = y y y y 3 = y ( ) (B) Uing eq B in the expreion for mean n and variance σ in eq 7, we find = ( ) n i = ϕ() σ = ( ) ( i) = (B) ϕ() ϕ() = 3 ϕ() (B3) The ratio of thi variance and mean give the Fano factor defined in eq 5. For the MM cheme, uing eq A and eq 5, the Fano factor i given by αa [S] F = ([S] + K ) A (B4) where α A =k /k and K A =(k + k E )/k. Thi i identical to F obtained previouly by olving the mater equation for the generating function. 3, AUTHOR IFORMATIO ote The author declare no competing financial interet. ACKOWLEDGMETS The proect wa upported by Grant o. P-RR8754 from the ational Center for Reearch Reource (CRR), a component of the ational Intitute of Health (IH). The work at LAL wa carried out under the aupice of the ational uclear Security Adminitration of the U.S. Department of Energy at Lo Alamo ational Laboratory under Contract o. DE-AC5-6A5396. The work at MC wa additionally upported by SF under Grant o. ECCS Jianhu Cao acknowledge the upport by ARO DOD (W9F-9-488) and SMART through ID-IRG. REFERECES () Michaeli, L.; Menten., M. L. Biochem. Z. 93, 49, 333. () Berezhkovkii, A. M.; Bezrukov, S. M. Chem. Phy. 5, 39, 34. (3) Bezrukov, S. M.; Berezhkovkii, A. M.; Szabo, A. J. Chem. Phy. 7, 7, 5. (4) Cao, J. J. Phy. Chem. B, 5, (5) Wu, J.; Cao, J. Adv. Phy. Chem., 46, 39. (6) Bezrukov, S. M.; Berezhkovkii, A. M.; Putovoit, M. A.; Szabo, A. J. Chem. Phy., 3, 86. (7) Berezhkovkii, A. M.; Putovoit, M. A.; Bezrukov, S. M. J. Chem. Phy., 6, 995. (8) Berezhkovkii, A. M.; Putovoit, M. A.; Bezrukov, S. M. J. Chem. Phy. 3, 9, (9) Kaianowicz, J. J.; Brandin, E.; Branton, D.; Deamer, D. W. Proc. atl. Acad. Sci. U.S.A. 996, 93, 377. () Mandel, L. Opt. Lett. 979, 4, 5. () Short, R.; Mandel, L. Phy. Rev. Lett. 983, 5, 384. () He, Y.; Barkai, E. J. Chem. Phy. 5,, (3) de Ronde, W. H.; Daniel, B. C.; Mugler, A.; Sinityn,. A.; emenman, I. IET Syt. Biol. 9, 3, 49. (4) Jung, W.; Yang, S.; Sung, J. J. Phy. Chem. B, 4, 984. (5) Cao, J.; Silbey, R. J. J. Phy. Chem. B 8,, 867. (6) Cox, D. R. Renewal Theory; Methuen & Co Ltd.: London, 96. (7) Cao, J. Chem. Phy. Lett., 37, 38. (8) Gopich, I. V.; Szabo, A. J. Chem. Phy. 3, 8, 454. (9) Gopich, I. V.; Szabo, A. J. Chem. Phy. 6, 4, 547. () Gardiner, C. W. Handbook of tochatic method: FOR Phyic, Chemitry, and the natural cience; Springer: ew York, 996. () Sinityn,. A.; emenman, I. Euro Phy. Lett. 7, 77, 58. () van Kampen,. G. Stochatic procee in phyic and chemitry; Elevier Science: ew York, 99. (3) Ferht, A. Structure and mechanim in protein cience; W. H. Freeman and Co.: ew York, 998. (4) Berg, J. M.; Tymoczko, J. L.; Stryer, L. Biochemitry, 5th ed.; W H Freeman: ew York,. (5) Hahn, H. S.; itzan, A.; Ortoleva, P.; Ro, J. Proc. atl. Acad. Sci. U.S.A. 974, 7, 467. (6) Haty, J.; Pradine, J.; Dolnik, M.; Collin, J. J. Proc atl Acad Sci U S A., 97, 75. (7) Haty, J.; Collin, J. J. at. Genet., 3, 3. (8) Ozbudak, E. M.; Thattai, M.; Kurter, I.; Groman, A. D.; van Oudenaarden, A. at. Genet., 3, 69. (9) Raer, J. M.; O Shea, E. K. Science 5, 39,. 59 dx.doi.org/./p J. Phy. Chem. B 3, 7, 53 59