An exactly solvable model for a beta-hairpin with random interactions
aa r X i v : . [ c ond - m a t . d i s - nn ] O c t An exactly solvable model for a β -hairpin withrandom interactions Marco Zamparo
Dipartimento di Fisica, INFN sezione di Torino and CNISM,Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, ItalyE-mail: [email protected]
Abstract.
I investigate a disordered version of a simplified model of protein folding,with binary degrees of freedom, applied to an ideal β -hairpin structure. Disorderis introduced by assuming that the contact energies are independent and identicallydistributed random variables. The equilibrium free-energy of the model is studied,performing the exact calculation of its quenched value and proving the self-averagingfeature. n exactly solvable model for a β -hairpin with random interactions
1. Introduction
The present paper is devoted to the analysis of a simple disordered model for an ideal β -hairpin structure, for which some exact results may be derived. Disordered modelsoriginate very intricated scenario and their study needs new mathematical methods andalgorithms; reffering to plain models with a known solution could be helpful to testthem.The model I consider is a disordered version of one introduced by Wako and Saitˆo[1, 2] in 1978 and independently reintroduced by Mu˜noz and co-workers [3, 4, 5] in thelate 90’s to inquire into the problem of protein folding. The Wako-Saitˆo-Mu˜noz-Eaton(WSME) model is a highly simplified one where the purpose is describing the equilibriumof the protein folding process under the assumption that it is mainly determined bythe structure of the native state (the functional state of a protein), whose knowledge isassumed. It is a one-dimensional model, with long-range, many-body interactions, wherea binary variable is associated to each peptide bond (the bond connecting consecutiveaminoacids), denoting the native and unfolded conformation. Two aminoacids caninteract only if they are in contact in the native state and all the peptide bonds betweenthem are ordered. Moreover an entropic cost is associated with each ordered bond.Many papers have been published in the last few years concerning the equilibriumproperties of the model and its exact solution [6, 7, 8], its kinetics [9, 10, 11] and somegeneralizations to the problem of mechanical unfolding [12, 13]. In particular in [6] theexact solution for a homogeneous β -hairpin structure was given, while in [7] one canfind the exact treatment in the general case. Recently the model has been applied tothe analysis of real proteins [14, 15, 16, 17, 18, 19, 20, 21] and, rather interestingly, ina problem of strained epitaxy [22, 23, 24].In order to introduce some disorder in the WSME model, I suppose the contactenergies are independent quenched variables. This assumption has been done for thebase pairing energies in some models for the ribonucleic acid (RNA) secondary structure[25], where one aims at retaining the spirit of Watson-Crick pairing that interactionsbetween some specific bases are favoured with respect to the others. However, evenif the β -hairpin structure mimics the zipper features of the RNA secondary structure,the purpose of this paper is the modest one of proposing a simple exactly solvabledisordered model, calculating the free-energy and proving its self-averaging property.The computation of the quenched free-energy, i.e. the average of the free-energy overthe quenched disorder, will be provided avoiding the replica theory [26] and making useof some properties of the free-energy itself which will be proven rigorously in advance.The paper is organised as follow: in Section 2 the WSME model and its disorderedversion for the β -hairpin structure are introduced. Section 3 is devoted to the calculationof the quenched free-energy and Section 4 to prove self-averaging. Conclusions are drawnin Section 5. n exactly solvable model for a β -hairpin with random interactions
2. The model
The WSME model describes a protein of N + 1 residues as a chain of N peptidebonds connecting consecutive aminoacids. In order to identify the native (ordered)conformation and distinguish it from the unfolded (disordered) one, a binary variable m k is associated to the peptide bond k , k = 1 , . . . , N . Each variable, related to the valuesof the dihedral angles at the same peptide bond, assumes value 1 in the native stateand 0 otherwise. Since the unfolded state allows a much larger number of microscopicrealizations than the native one, an entropic cost q k is given to the ordering of the peptidebond k . The main assumption about the interactions is that two bonds can interactonly if they are in contact in the native state (so that the model can be classified asG¯o-like [27]) and all bonds between them are ordered.The Hamiltonian of the model (an effective free-energy, properly speaking) reads H N ( m ) = N − X i =1 N X j = i +1 ǫ ij ∆ ij j Y k = i m k + k B T N X k =1 q k m k , (1)where T is the absolute temperature. The product Q jk = i m k takes value 1 if and onlyif all the peptide bonds going from i to j are ordered, thereby realizing the assumedinteraction. The contact matrix elements ∆ ij ∈ { , } tell us which bonds are at closedistance in the native state. Finally, the contact energies ǫ ij < β -hairpin with an odd number 2 N + 1 of peptide bonds is characterizedby the contact matrix elements ∆ ij equal to 1 if i + j = 2 N + 2 and 0 otherwise. Thestructure results in the characteristic Hamiltonian (divided by k B T ) H ǫN ( m ) = − β N X i =1 ǫ i N +1+ i Y k = N +1 − i m k + q N +1 X k =1 m k , (2)where β = 1 /k B T .In this work I concentrate on the case in which ǫ , . . . , ǫ N are independent randomvariables identically distributed in a set E ⊆ R according to a probability measure P .Moreover, in order to deal with a homogeneous model having a thermodynamic limit,the entropic cost q k is chosen equal to q for any k , as the comparison between theHamiltonians (1) and (2) shows. I shall assume P is any probability measure satisfyingthe condition R ǫ ∈E exp ( βǫ ) P ( dǫ ) < ∞ , given an arbitrary real value of β , and from nowon I will denote by µ the expectation of the contact energy and with P N the productmeasure P × . . . × P N -times.Let us denote with f N the quenched free-energy (times β ) f N ( β, q ) = − N + 1 E [log Z N ] . = − N + 1 Z ǫ ∈E N log Z N ( ǫ ) P N ( dǫ ) , (3)where Z N ( ǫ ) is the partition function of the model (2) given a sequence ǫ = ( ǫ , . . . , ǫ N )of interaction energies: Z N ( ǫ ) = X m ∈{ , } N +1 exp[ − H ǫN ( m )] . (4) n exactly solvable model for a β -hairpin with random interactions f . = lim N →∞ f N is the quenched free-energy in the thermodynamic limit.
3. The free-energy
In this section I show how to compute exactly the quenched free energy, discussing someof its properties in advance and then exploiting them to perform the calculation. Let usstart by observing that, due to the features of the model, it is possible to simplify theexpression of the partition function Z N . Indeed, summing over the binary variables m and m N +1 allows to find the iterative equation [6] Z N ( ǫ ) = (1 + e − q ) Z N − ( ǫ ) + (e βǫ N − P N − i =1 ǫ i − q (2 N +1) (5)valid for any N ∈ N . Joining this relation to the initial condition Z ( ǫ ) = (1 + e − q ) + (e βǫ − − q , (6)one obtains immediately the expression Z N ( ǫ ) = (1 + e − q ) N +1 ++ N X n =1 (e βǫ n − β P n − i =1 ǫ i − q (2 n +1) (1 + e − q ) N − n ) . (7)The formula for Z N can still be slightly reduced, as it is stated by the followingproposition. Proposition 1.
There exist two positive constants with respect to N , C and D , suchthat C " N X n =1 e β P ni =1 ǫ i (1 + e q ) n ≤ Z N ( ǫ )(1 + e − q ) N +1 ≤ D " N X n =1 e β P ni =1 ǫ i (1 + e q ) n . (8)Before sketching the proof, in order to deal with more compact formulas in thefollowing, it is convenient to introduce the new quantitiesΞ β,λN ( ǫ ) = 1 + N X n =1 e β P ni =1 ǫ i − λn (9)and g N ( β, λ ) = 1 N E [log Ξ β,λN ] (10)where the explicit dipendence on β and λ is taken into account, and rewrite f in theform f ( β, q ) = − log(1 + e − q ) − g ( β, q )) (11)with g . = lim N →∞ g N . The relationship between the free-energy and the modelparameters comes from the evaluation of the function g , so that I shall focus on g rather than f . n exactly solvable model for a β -hairpin with random interactions Proof of Proposition 1.
Looking at the expression (7) and splitting the term (e βǫ n − Z N in the following manner: Z N ( ǫ )(1 + e − q ) N +1 = 1 − (1 + e q ) − ++ 1 − (1 + e q ) − q N − X n =1 e β P ni =1 ǫ i (1 + e q ) n + (1 + e q ) − e β P Ni =1 ǫ i (1 + e q ) N . (12)The statement of the proposition is achieved by choosing C = min n − (1 + e q ) − , − (1 + e q ) − q , (1 + e q ) − o > D = max n − (1 + e q ) − , − (1 + e q ) − q , (1 + e q ) − o > . (14)Let us now go over the properties of g that shall allow its evaluation. From aphysical point of view one is interested only in positive values of β and λ , but foranalitycal reasons it is convenient to assume β and λ taking any real value. The firstproperty I show concerns the behaviour of g under reflection with respect to the origin. Proposition 2. g ( β, λ ) = βµ − λ + g ( − β, − λ ) where µ is the expectation value of theenergy contact: µ = Z ǫ ∈E ǫP ( dǫ ) . (15) Proof of Proposition 2.
Remembering the definition (9), we haveΞ β,λN ( ǫ ) = 1 + N − X n =1 e β P ni =1 ǫ i − λn + e β P Ni =1 ǫ i − λN = e β P Ni =1 ǫ i − λN " e − β P Ni =1 ǫ i + λN + N − X n =1 e − β P Ni = n +1 ǫ i + λ ( N − n ) + 1 (16)and changing n with N − n in the sum, we can go on writingΞ β,λN ( ǫ , . . . , ǫ N ) = e β P Ni =1 ǫ i − λN " e − β P Ni =1 ǫ i + λN + N − X n =1 e − β P Ni = N − n +1 ǫ i + λn + 1 = e β P Ni =1 ǫ i − λN " N X n =1 e − β P Ni = N − n +1 ǫ i + λn = e β P Ni =1 ǫ i − λN " N X n =1 e − β P ni =1 ǫ N − i +1 + λn = e β P Ni =1 ǫ i − λN Ξ − β, − λN ( ǫ N , . . . , ǫ ) . (17)The connection (10) between Ξ β,λN and g N allows us to conclude immediately the proof.The second result I report describes a homogeneity property of g . Proposition 3. g ( tβ, tλ ) = tg ( β, λ ) for any t > n exactly solvable model for a β -hairpin with random interactions Proof of Propostion 3.
At first let us suppose t ≥
1. From the inequality, valid for x >
0, (1 + x ) t ≥ x t (18)and from the convexity of the function x → x t , x >
0, it follows that N X n =0 a tn ≤ N X n =0 a n ! t ≤ ( N + 1) t − N X n =0 a tn (19)for any integer N and positive numbers a , . . . , a N . This chain of inequalities impliesΞ tβ,tλN ( ǫ ) ≤ h Ξ β,λN ( ǫ ) i t ≤ ( N + 1) t − Ξ tβ,tλN ( ǫ ) (20)and then g ( tβ, tλ ) = tg ( β, λ ) when t ≥
1. Bearing in mind the latter point, thesubstitution of β with β/t and λ with λ/t allows us to prove the proposition also when0 < t < g in a region of the parameter space. Proposition 4. g ( β, λ ) = 0 if λ ≥ log R ǫ ∈E e βǫ P ( dǫ ). Proof of Proposition 4.
Making use of the concavity of the logarithm function, weobtain 0 ≤ g N ( β, λ ) ≤ N log Z ǫ ∈E N Ξ β,λN ( ǫ ) P N ( dǫ )= 1 N log " N X n =1 (cid:18) e − λ Z ǫ ∈E e βǫ P ( dǫ ) (cid:19) n . (21)Then g ( β, λ ) = 0 if e − λ R ǫ ∈E e βǫ P ( dǫ ) ≤ λ ≥ log R ǫ ∈E e βǫ P ( dǫ ).Exploiting these properties, it is now feasible to show the form of the function g for the whole parameter space. From proposition 3 and 4 it follows that, given t largerthan 0, g vanishes if λ ≥ t log R ǫ ∈E e tβǫ P ( dǫ ). Taking the limit t → + , this conditionreduces to λ ≥ βµ . On the other hand, if λ ≤ βµ then − λ ≥ − βµ and proposition 2tells us that g ( β, λ ) = βµ − λ , due to the null value of g ( − β, − λ ). Let us conclude bycollecting the previous results in a compact formula by means of the Heaviside function θ ( θ ( x ) = 1 if x ≥ x ) = xθ ( x ). The followingholds Theorem 1. g ( β, λ ) = ( βµ − λ ) θ ( βµ − λ ) = Θ( βµ − λ ).
4. Self-averaging property
This section is devoted to the proof of the self-averaging feature of the free-energy. Inorder to quantify the fluctuations of the free-energy let us introduce the function S N defined as S N ( β, λ ) = E h(cid:12)(cid:12)(cid:12) N log Ξ β,λN − g ( β, λ ) (cid:12)(cid:12)(cid:12)i . (22) n exactly solvable model for a β -hairpin with random interactions δ , the probability of having a fluctuationlarger than or equal to δ is bounded by S N : P h(cid:12)(cid:12)(cid:12) N log Ξ β,λN − g ( β, λ ) (cid:12)(cid:12)(cid:12) ≥ δ i ≤ S N ( β, λ ) δ , (23)where the left-hand side is an usual short notation denoting the probability measure ofthe set of ǫ ∈ E N such that g ( β, λ ) − δ ≤ N log Ξ β,λN ( ǫ ) ≤ g ( β, λ ) + δ .The self-averaging property of the free-energy is described by the fact that S N vanishes in the thermodynamic limit, as the following theorem states Theorem 2. S ( β, λ ) . = lim N →∞ S N ( β, λ ) = 0 for any real numbers β and λ .In order to prove the theorem it is useful to extend to S the reflection result about g . Proposition 5. S ( β, λ ) = S ( − β, − λ ). Proof of Proposition 5.
From relation (17) and proposition 2 we have1 N log Ξ β,λN ( ǫ , . . . , ǫ N ) − g ( β, λ ) = β (cid:16) N N X i =1 ǫ i − µ (cid:17) ++ 1 N log Ξ − β, − λN ( ǫ N , . . . , ǫ ) − g ( − β, − λ ) , (24)which, passing to absolute values and averaging, yields (cid:12)(cid:12)(cid:12) S N ( β, λ ) − S N ( − β, − λ ) (cid:12)(cid:12)(cid:12) ≤ | β | Z ǫ ∈E N (cid:12)(cid:12)(cid:12) N N X i =1 ǫ i − µ (cid:12)(cid:12)(cid:12) P N ( dǫ ) . (25)Thanks to the Cauchy-Schwarz inequality we can go on and reach the result (cid:12)(cid:12)(cid:12) S N ( β, λ ) − S N ( − β, − λ ) (cid:12)(cid:12)(cid:12) ≤ | β | vuutZ ǫ ∈E N (cid:16) N N X i =1 ǫ i − µ (cid:17) P N ( dǫ )= | β |√ N sZ ǫ ∈E ( ǫ − µ ) P ( dǫ ) . (26)The proof is concluded considering the limit N → ∞ .Now we can come back to the theorem. Proof of Theorem 2.
Remembering that g ( β, λ ) = 0 if βµ ≤ λ and observing thatΞ β,λN ( ǫ ) ≥
1, we have, when βµ ≤ λ , S N ( β, λ ) = E h(cid:12)(cid:12)(cid:12) N log Ξ β,λN (cid:12)(cid:12)(cid:12)i = E h N log Ξ β,λN i = g N ( β, λ ) (27)and then lim N →∞ S N ( β, λ ) = g ( β, λ ) = 0. On the other hand, when βµ > λ we obtainfrom proposition 5 that S ( β, λ ) = S ( − β, − λ ) = 0 since − βµ < − λ .
5. Conclusions
In the previous sections we focused on the function g , since its study was equivalent tothat of the free-energy f . Now we can come back to the expression (11) and thanks tothe theorem 1 write the final formula f ( β, q ) = − log(1 + e − q ) −
12 Θ( βµ − q )) . (28) n exactly solvable model for a β -hairpin with random interactions g and thus its behaviour iscompletely characterized.The continuous function Θ( x ) has a discontinuity in the first derivative at x = 0showing that a first order phase transition occurs at the critical value β c ( q ) = µ log(1+e q )of β . This critical point is associated to the transition between a disordered phase, theunfolded state of the peptide, and an ordered one, the native state, pointing out atwo-state behaviour.The transition can be better characterized by means of an order parameter p N ,function of β and q , measuring the level of the order in the system. We can choose p N as the thermal and then quenched average of the fraction of native bonds. Fromdefinitions (2), (3) and (4) it follows the result p N ( β, q ) = ∂f N ( β, q ) ∂q (29)which, passing to the limit N → ∞ , allows us to obtain p ( β, q ) . = lim N →∞ p N ( β, q ) = β > β c ( q ),11 + e q if β < β c ( q ). (30)At low temperature, β > β c ( q ), all the peptide bonds are ordered and the protein isin its native state. The relationship between β c and the expectation contact energy µ implies that no ordering can occur at physical temperature when the interaction isrepulsive in average ( µ < µ . This means that the quenched disorder doesnot affect the critical behaviour and the transition remains sharp of the first order,as in the pure case. This feature could not be considered manifest a priori , since, asfar I know, no general result is available for models with long-range and many-bodyinteractions in the presence of quenched disorder.Concluding, in this paper I have studied and solved exactly a simple disorderedmodel, showing at first the mathematical expression of the quenched free-energy andthen characterising completely the distribution of the free-energy by proving its self-average feature. The replica trick has been avoided since a more straightforward wayhas been found to reach the desired results. I believe these might turn out to be helpful asa benchmark for testing methods from disordered system theory, where exact solutionsare quite rare. Acknowledgments
I am grateful to Alessandro Pelizzola for having stimulated me to produce this work.
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