Embedding a Native State into a Random Heteropolymer Model: The Dynamic Approach

21 Pages·0.34 MB·English

by  Zoran Konkoli

Embedding a Native State into a Random
Heteropolymer Model: The Dynamic Approach
Z. Konkoli1,2 and J. Hertz2
3 1Department of Applied Physics,
0 Chalmers University of Technology and G¨oteborg University,
0 SE 412 96 G¨oteborg, Sweden
2 2NORDITA, Blegdamsvej 17, DK 2100 København, Denmark

n
(February 1, 2008)
a
J
WestudyarandomheteropolymermodelwithLangevindynamics,inthesupersymmetricformu-
4
lation. Employingaproceduresimilartoonethathasbeenusedinstaticcalculations, weconstruct
2
an ensemble in which the affinity of the system for a native state is controlled by a “selection tem-

] perature” T0. In the limit of high T0, the model reduces to a random heteropolymer, while for
h T0 → 0 the system is forced into the native state. Within the Gaussian variational approach that
c weemployedpreviouslyfortherandomheteropolymer,weexplorethephasesofthesystemforhigh
e
m andlowT0. ForhighT0,thesystemexhibitsa(dynamical)spinglass phase,likethatfoundforthe
randomheteropolymer,belowatemperatureTg. ForlowT0,wefindanorderedphase,characterized
t- by a nonzero overlap with the native state, below a temperature Tn ∝ 1/T0 > Tg. However, the
a random-globulephaseremainslocallystablebelowTn,downtothedynamicalglasstransitionatTg.
t
s Thus, inthis model, folding israpid for temperaturesbetween Tg and Tn,butbelow Tg thesystem
. can get trapped in conformations uncorrelated with the native state. At a lower temperature, the
t
a ordered phase can also undergo a dynamical glass transition, splitting into substates separated by
m
large barriers.
-
d 05.70.Ln, 87.14.Ee
n
o
c I. INTRODUCTION strains the motion of the system drastically, and it can-
[
not explore its full configuration space and reach Gibbs

2 equilibrium. In a previous paper ( [22], henceforth re-
v Theproteinfoldingprocessisrelevantforallaspectsof ferredtoaspaperI),wedemonstrated,inmeanfieldthe-
6 life: once readofffrom the RNA chain,proteins perform ory, the existence of a sharp transition to a “dynamical
8
avarietyoffunctions,frommechanicalworktoattacking glassystate”inwhichtheequilibrationtimedivergesand
2
viruses.[1]Thekeyfactorwhichdeterminesthe function the dynamics exhibit aging. (The potential importance
7
0 of a protein molecule is its 3D structure, which, in turn, of spin glass physics to proteins was first discussed in
2 isdeterminedbythesequenceofaminoacidsformingthe Ref. [23]). Obviously, the random heteropolymer model
0 proteinchain.[2–5]Furthermore,aproteinthathasbeen does not describe a protein with a native state, but it
t/ denatured (by stretching it for example) finds its native alerts us to the need to examine possible glassiness in
a state relatively quickly. Protein folding has attracted an models for protein dynamics.
m
enormousamountofscientificattention,butstillthereis
Why are real proteins not glassy? Evidently, nature
- no generic understanding of this process. Nevertheless,
d has tuned amino acid sequences to avoid glassy behav-
one thing is clear: a proteins generally has a potential
n ior. To understand how such tuning might be done, it
o energy surface which results in a stable free energy min- is worthwhile to study models which contain competi-
c imum, corresponding to the native state [3]. tion between glassiness and a tendency to form a native
:
v Randomheteropolymermodels(RHP)havebeenused state, by choosing interactions which are not completely
i extensively as candidate systems which might help us random. Severalstudiesalongthelinesofthissuggestion
X
to understand the generic features of the potential en- have been made in statics (using the replica treatment,
r
a ergy surfaces of proteins and their connection with ther- see, e.g., Ref. [23]). The tendency toward a particular
modynamic [6–13] and dynamical [14–20] properties. state can be built in by choosing sequences from a dis-
The RHP model is characterized by quenched random tribution correlated with the native sequence [2,24–26].
monomer-monomerinteractions,meantto mimic the va- A dynamical treatmentof similar models is highly desir-
riety of interactions between amino-acids in random se- able, not only to help gain insight into results obtained
quences. It turns out that the potential energy surface in replica approaches,but also because knowledge of the
of the RHP is quite similar to that of a particular class correctthermodynamicsalonemaynotbesufficient: itis
ofspinglasses[21]: Itscomplexform,withexponentially knownthatinrelated(meanfieldmodels)static anddy-
large numbers of local minima and saddle points, con- namicphasediagramscanbedifferent. Thus(atleaston
1
sufficiently shorttime scales)only adynamicalapproach dergo a dynamical freezing into a different glassy phase.
candescribe the measurablepropertiesofthe system. In In this phase the conformation of the protein is always
this paper we undertake such a study. highly correlated with the native state, but cooperative
kinetic constraints still lead to a divergent equilibration
We extend the RHP model studied in [6,7] to include
time, as for the frozen-globule state.
the existence of a native state: the original random
monomer-monomerinteractionsare biasedsoas to favor
the native state conformation. The problem is formu-
latedasaLangevinmodel. Tothebestofourknowledge, II. THE MODEL
there is so far neither a static nor a dynamic treatment
availableforamodelofthissort: Staticstudieshavebeen
Themodelisdefinedasfollows. TheLangevindynam-
based on random monomer sequences, i.e., using only N
ics is assumed to be governed by a Hamiltonian H[x],
random parameters, see Refs. [2,24–26], rather than the
N(N 1)/2 in the RHP model.
− ∂x(s,t)/∂t= δH[x]/δx(s,t)+η(s,t). (1)
−
Admittedly, the model does not describe a realistic
protein (e.g., it does not give rise to secondary structure Here x(s,t) is the position of monomer s at time t. The
such as α-helices or β-sheets). However, it does con- monomers are numbered continuously from s = 0 to
tain important generic features: the polymeric structure s=N. η(s,t) is Gaussian noise
and the mixture of attractive and repulsive interactions.
Together, these features lead to frustration in the struc- η(s,t)η(s,t) =2Tδ(s s)δ(t t), (2)
′ ′ T ′ ′
h i − −
tural dynamics. In our view, ours is the simplest such
model that includes competition between glassy and na- resultingfromcouplingtoaheatbathattemperatureT.
tive states. As we willsee,it teachesus thatone cannot
The Hamiltonian H[x] contains a deterministic part
get rid of glassiness so easily.
H [x,µ] and a random part H[x, B ]. H [x,µ] is de-
0 0
As in paper I, we simplify the model further by omit- { }
fined as
ting three-body interactions in the polymer. (A re-
viewdescribinghowtoinclude three-bodytermsisgiven T N
H [x,µ]= ds [∂x(s,t)/∂s]2+µx(s,t)2 . (3)
in [9].) The price we have to pay for this simplification 0 2 { }
Z0
is that we have to introduce a somewhat arbitrary con-
fining potential,whichwetaketo haveaquadraticform.
Itdescribesthe elasticpropertiesofthe chainandacon-
We adjust its strength so that the radius of gyration R
g finement potential which fixes the density of the pro-
of a polymer of size N scales like N−1/d, where d is the tein. The radius of gyration Rg µ−1/4, so, in order
dimensionality of the system. In this way we attempt to that the protein is compact, i.e., R∼ N1/d, we require
g
describe a globularstate. Ofcourse,we cannot describe ∼
µ N 4/d. Thus, since we are interested in very long
−
the θ-point transition in such a model, but here we are ∼
proteins(to obtainthethermodynamiclimit)weneedto
only interested in transitions between different globular
solve the model for µ close to zero.
states.
The random part H[x, B ] describes the quenched
Our formal starting point is the Martin-Siggia-Rose { }
random interactions between monomers,
generating functional for the Langevin dynamics of the
model[27–30],written,forconvenienceandcompactness, 1 N
in its supersymmetric form [31]. To derive equations H[x, B ]= dsds′Bss′V(x(s,t) x(s′,t)). (4)
{ } 2 −
of motion for correlation and response functions we use Z0
a variational ansatz with a quadratic action. This ap-
proachhasbeenusedtostudy theproblemofamanifold We take Bss′ Gaussian, with variance B2. The
inarandompotential,inbothstatics[32,33]anddynam- quenched average over Bss′ is performed as (.) B =
h i
ics [34,35]. s>s′dBss′(.)P({B}). V(∆x) is a short-range poten-
tial, and, for simplicity, we take it to have a Gaussian
In paper I we showed that the RHP model exhibited RforQm, as in Ref. [16],
broken ergodicity (formally, a spontaneous supersym-
mpheatsrey.brIenaktihneg)pirnesaenltowst-tuedmy,pewraitthurientdeyrnaactmioicnaslbgilaassesdy V(∆x)= 1 d/2e−(∆x)2/2σ. (5)
2πσ
in favor of a native state to a controlled degree, we (cid:18) (cid:19)
find, in addition, a well-folded phase, if the bias is
d is the dimensionality of the system and √σ the range
strong enough. It can coexist with either the disordered
of the potential. Large (small) σ corresponds to a
(random-globule)stateorthe frozen-globuleglassphase,
long (short) range potential. In particular, for σ 0,
depending on the temperature. Furthermore, we find
→
V(∆x) δ(∆x), and we recover the potential used
that at low temperature the native phase can itself un-
→
in [6,7,18]. Here and in the following ∆x refers to a
2
monomer-monomer distance: ∆x = x(s,t) x(s′,t) for [Φ] = DΦ [Φ]e S[Φ], (8)
− T −
a pair of monomers s, s. hO i O
′ Z
S[Φ]=S [Φ]+S[Φ,x , B ], (9)
We use reasoning similar to that employed in statics 1 0 { }
todefine P( B )(see Refs.[2,24–26]),adaptingittothe
random-bon{d m}odel: where
P( B ) e−T10H[x0,B]−12 dsds′Bs2s′/2B2 (6) S1[Φ]=1/2 dsd1ds′d2Φ(s,1)K1s2s′Φ(s′2), (10)
{ } ∝ Z
R
S[Φ,x , B ]=1/2 d1dsds
T is called the selection temperature, and x (s) is some 0 { } ′×
0 0
Z
arbitrarynativestateconformation. Thusthesymmetric Bs,s′V(Φ(s,1) Φ(s′,1)), (11)
bonddistribution ofthe RHP model is distortedsoas to × −
givebiggerweightto Bss′’swhichareattractivebetween and
monomers which lie close to each other in the configura-
tion x0(s). Explicitly, the properly normalized P({B}) K1s2s′≡δ12δss′K1s , K1s =T µ−(∂/∂s)2 −D1(2), (12)
is given by ∂2 ∂2 ∂
D(2)=2T +2θ (cid:2) . (cid:3) (13)
P( B )=(2πB2)−N(N−1)/4e−β02B2/4 dsds′V(x0(s)−x0(s′))2 1 ∂θ1∂θ¯1 1∂θ1∂t1 − ∂t1
{ }
e−β0/2 dsds′Bss′V(x0(s)−x0(s′))−1/2 Rdsds′Bs2s′/2B2, (7) The Φ(s,1) denotes a superfield
×
R R Φ(s,1)=x(s,t )+θ¯ η(s,t )+
1 1 1
fromwhichweseethatthedistributionofBss′ ispeaked +η¯(s,t1)θ1+θ¯1θ1x˜(s,t1) (14)
around Bmax = β B2V(x (s) x (s)). Thus, if
ss′ − 0 0 − 0 ′
monomerssands arecloseinthenativestate(V(x (s)
′ 0 containingthe physicalcoordinatex(s,t), the MSR aux-
−
x0(s′))=0),theircouplingconstantBss′ ispulleddown, iliary field x˜(s,t), ghost fields η(s,t) and η¯(s,t) that en-
6
as in a Go model [36,37]. For T we recover the
0 force the normalization of the distribution, and Grass-
→ ∞
RHP model. For T0 0, P( B ) picks a specific set mann variables θ and θ¯. We use the notation 1
→ { }
of Bss′. For this set, by construction, x0(s) is the deep- (θ ,θ¯ ,t ), likewise d1 dθ¯ dθ dt . ≡
1 1 1 1 1 1
est minimum of H[x, B ] given in Eq. (4). This is the ≡
mechanism that embe{ds}the native state x (s). Of course, the soRlution cRan be obtained without the
0
aidofthe supersymmetric formalism,but we find it con-
This mechanism is somewhat arbitrary. However,
veniently compact.
the fact that the strength of embedding of the native
As noticed by De Dominicis [28] the expression in
state is controlled by the single parameter T facilitates
0
Eq.(8) is already normalized, so the average over the
the study of transitions between random and native-like
states (and, as we will show, of possible coexistence of quenched disorder Bs,s′ can be done directly on (8):
such phases).
[Φ] = DΦ [Φ]e (S1[Φ]+S2[Φ,x0]), (15)
Sofar,theconfigurationx0(s)isarbitrary. Thusx0(s) hhO iTiB O −
Z
has to be considered a quenched random function, to be
averaged over just like Bss′ in order to obtain generic where exp( S2[Φ,x0]) exp( S[Φ,x0, B ]) B, and
− ≡h − { } i
results. We will carry this average out later.
B2 2
All our results are obtained in the thermodynamic S [Φ,x ]= dsds d1V(Φ(s,1) Φ(s,1))
2 0 ′ ′
− 4 −
limit, where the length N of the heteropolymer chain Z (cid:20)Z (cid:21)
goes to infinity. Also, for simplicity, we join the polymer β0B2
dsdsd1V(Φ(s,1) Φ(s,1))
ends to form a ring. This neglect of end effects is valid − 2 ′ − ′ ×
Z
for a long chain.
V(x (s) x (s)). (16)
0 0 ′
× −
Thus, the native state x (s) enters the action in the
0
III. MAPPING TO THE FIELD THEORY second term of Eq. (16). Note that there is no term
β2V(x (s) x (s))2, since it gets cancelled by a similar
0 0 − 0 ′
normalization factor for P( B ) in Eq. (7). It is useful
{ }
To solve the model we map the Langevin dynamics to rewrite Eq. (16) as
onto a supersymmetric (SUSY) field theory. Using the
B2
standard Martin-Siggia-Rose formalism [27–30] and su- S = ddxddyd1d2A(V)(x,y)A(δ)(x,y)
persymmetric(SUSY)notation[19,20,31,38],thedynam- 2 − 4 12 12
Z
ical averageof any observable, for fixed B , can be cal- β B2
{ } 0 ddxddyd1A(V)(x,y)A(δ)(x,y) (17)
culated as (see, e.g., Paper I for details), − 2 10 10
Z
3
with the notation A(f)(x,y) = dsf(Φ(s,1) There is some formal similarity between the dynamical
12 −
x)f(Φ(s,2) y),A(f)(x,y)= dsf(Φ(s,1) x)f(x (s) functional Fdyn and the static replica partition function.
− 10 R − 0 − The integration over Dx enters in the same way as the
y); f V,δ . In the long-chain limit, as discussed 0
∈ { } R extra replica in the static formalism.
in Paper I (and references therein), one obtains a self-
consistent field theoretic formulation, with S simplified
2
to,
V. CORRELATION FUNCTIONS
B2
S [Φ,x ]= ddxddyd1d2 A(V)(x,y) A(δ)(x,y)
2 0 4 h 12 ih 12 i−
Z h The SUSY correlationfunctions
−A1(V2)(x,y)hA1(δ2)(x,y)i−hA(1V2)(x,y)iA1(δ2)(x,y) Gss′ Φ(s,1)Φ(s,2) (24)
+β02B2 ddxddyd1 hA(1V0)(x,y)ihA1(δ0)(x,y)i i Gs11s02′ ≡≡hhΦ(s,1)x0(s′′)ii (25)
A(V)(xZ,y) A(δ)(x,hy) A(V)(x,y) A(δ)(x,y) . (18) Gs0s0′ ≡hx0(s)x0(s′)i (26)
− 10 h 10 i−h 10 i 10
i contain all the information we are interested in.
All averages of the type A(V,δ) have to be calculated Gss′ encodes16correlationfunctions,outofwhichonly
12
h i
self-consistently with S[Φ] = S [Φ]+S [Φ]. (We have two, correlation and response function, are independent
1 2
abbreviated the double average . simply by . .) and nonzero:
T B
hhi i hi
IdnestchreipltiimonitoNf t→he∞dynEaqms.ic(s15fo)rananda(1rb8)itrparroyvindaetaivneesxtaactet Gs1s2′ =C(s,t1;s′,t2)+(θ¯2−θ¯1)×
x (s). [θ2R(s,t1;s′,t2) θ1R(s′,t2;s,t1)], (27)
0 × −
with
C(s,t;s,t) x(s,t)x(s ,t) , (28)
IV. AVERAGE OVER NATIVE STATE ′ ′ ′ ′
≡h i
CONFORMATIONS δ x(s,t)
R(s,t;s′,t′) x(s,t)x˜(s′,t′) = h i. (29)
≡h i δh(s,t)
′ ′
It is impossible to solve the model for a general na-
The field h(s,t) entering the description of response
′ ′
tive state configuration x (s). We therefore consider a
0 function is an arbitrary external field that couples to
distribution of native states and perform the average
x(s,t). Thefactthatonlytwocorrelationfunctionssur-
′ ′
vive is related to Ward identities originatingfrom SUSY
<O[Φ,x0]>= Dx0 <O[Φ,x0]>e−S0[x0], (19) invariance of the original action S.
Z
The supersymmetry of the theory is associated with
where S [x ] weights each native state conformation in equilibrium. One of the Ward identities resulting
0 0
the ensemble as fromSUSYisthefluctuation-dissipationtheorem(FDT)
which relates correlation and response functions. In the
S [x ]=1/2 dsx (s)Kss′x (s), (20) presentcase,theglassystatemanifestsitselfasasponta-
0 0 0 00 0 ′
neous breaking of supersymmetry, leading to a modified
Z
FDT, as in previous treatments of other models [31,34].
with
Gss′ describes the overlap with the native state. Due
K0s0s′ ≡δss′(µ0−∂2/∂s′2). (21) toW1a0rdidentities,onlyasinglecorrelationfunctionsur-
vives (see Appendix A for details):
The parameter µ0 fixes a size of the globule in this en- Gss′ = x(s,t)x (s) φ(s,t ;s). (30)
semble, 10 h 0 ′ i≡ 1 ′
Similarly, the native state ensemble is described by
1
x (s)2 = (22)
h 0 i 2√µ0 Gs0s0′ =hx0(s)x0(s′)i≡Γ(s;s′). (31)
Since the polymer ends are joined, there is translational Gss′ alone is sufficient to describe the RHP model. Here
12
invariance along the coordinate s and hx0(s)2i does not we need the two extra functions Gs1s0′ and Gs0s0′.
depend on s. Thus, with this procedure, the dynamical
Also, in what follows, we exploit the translational
generating functional for the problem is calculated as
invariance along the s coordinate and define Fourier
transforms of all correlation functions: X(s,s) =
e−Fdyn = Dx0DΦe−(S0[x0]+S1[Φ]+S2[Φ,x0]). (23) dkeik(s s′)X where X =C,R,φ,Γ. ′
2π − k
Z
R
4
VI. EQUATIONS OF MOTION Using (33), (32)and (9) gives the following expression
for F :
dyn
To solve the model we proceed by making a Gaussian
variationalansatz(GVA), assumingthat the fields Φ are Fdyn = d2 dsds′K0s0s′Gs0s0′ +
described by the approximate action Z
+d dsdsd1d2Kss′Gss′ dTrlnG
Svar = 21 d1dsd2ds′Φ(s,1)(G−1)s1s2′Φ(s′,2)+ B22Z ′ 12 12 − 2
Z ddxddyd1d2 A(V)(x,y) A(δ)(x,y)
+ d1dsds′Φ(s,1)(G−1)s1s0′x0(s′)+ − 4 Z h 12 ih 12 i
β B2
+1Z dsdsx (s)(G 1)ss′x (s). (32) − 02 ddxddyd1d2hA(1V0)(x,y)ihA1(δ0)(x,y)i, (37)
2 ′ 0 − 00 0 ′ Z
Z
where all averages are to be calculated using S (see
Technically, this implies the following approximation for var
Eq. 32). Performing averages, the fourth and fifth term
F :
dyn on the right hand side of (37) become
F S +F . (33)
dyn var var
≈h i Fd(y4)n =−2dN d1d2dsds′V (B1s2+B1s2′)/2 , (38)
where Z h i
F(5) = β0d d1d2dsds (Bs +Bs′)/2 , (39)
e−Fvar ≡ Dx0DΦe−Svar =e(d/2)TrlnG, (34) dyn − N Z ′Vh 10 10 i
Z
. var =eFvar Dx0DΦ(.)e−Svar. (35) where
hi
Z
Bs = [Φ(s,1) Φ(s,2)]2 =Gss +Gss 2Gss, (40)
The stationarity condition B1s2 =h[Φ(s,1)−x (s)]2 i=Gss11+Gss22−2Gss1,2 (41)
10 h − 0 i 11 00− 10
δF
dyn
=0 (36)
δGss′ and
12
translates into the equation of motion for Green’s func- (z)= B˜2(z+σ) d/2 , B˜2 = B2N(4π) d/2. (42)
tion Gss′ (see Eqs. 43-46). We have derived iden- V − d − 2 v −
12
tical equations of motion by using the approach of
Ref. [35], where standard field theoretic identities (e.g.
Performing the variational ansatz (i.e., evaluating
ΦδS/δΦ = 0 ) are used. It can be shown that for
h i Eq. 36) results in the following equations of motion:
quadratic S the two procedures give the same result.
var
We omit this analysis here to save space.
Inacorrespondingequilibriumproblem,thestationar- T(µ+k2)−D1(2) Gk12 =δ12+2 d3V′(B13)×
ityconditionisalsoanextremumconditionandprovides h i Z
(Gk Gk )+2β (B )(Gk Gk ), (43)
aboundonthefreeenergy. Here,sinceFdyn containsin- × 32− 12 0V′ 10 02− 12
tegrations over complex fields and Grassmann variables,
theGVAdoesnotgiveaboundonF . Nevertheless,it
dyn
isthefirststepinasystematicapproximationprocedure, T(µ+k2) D(2) Gk =2 d2 (B )
− 1 10 V′ 12 ×
as outlined in Appendix B. h i Z
The GVA has been applied to the problem of a mani- (Gk20−Gk10)+2β0V′(B10)(Gk00−Gk10), (44)
foldinarandompotential,inbothstatics[32,33]anddy-
namics[34,35]. Themethodisexactwhenthedimension-
ality of the manifold is infinite but is only approximate (µ +k2)Gk =2β d2 (B )(Gk Gk ), (45)
0 01 0 V′ 20 21− 01
for finite dimensionality. Nevertheless, even for rather Z
low dimensionality it has been shown to be a very good
approximation in the random-manifold problem, where
it has been checked numerically [35]. We have shown in (µ +k2)Gk =1+2β d1 (B )(Gk Gk ), (46)
0 00 0 V′ 10 10− 00
Paper I that the present model is closely related to the Z
random-manifoldproblem. Thus,wehopethatthe GVA
willalsobereasonablehere,althoughwehavenotstrictly and after disentangling the SUSY notation one gets (see
checked its validity. Paper I for related details)
5
[T(µ+k2)+∂/∂t]C (t,t)= and, in the aging regime,
k ′
t
2TRk(t′,t)+2 dt′′ ′[B(t,t′′)]Rk(t′,t′′) lim C (t,λt)=q Cˆ (λ), (54)
V k k k
Z0 t
t →∞ 1
+4 dt′′V′′[B(t,t′′)]r(t,t′′)[Ck(t,t′)−Ck(t′′,t′)] tlim Rk(t,λt)= tRˆk(λ). (55)
Z0 →∞
2β0 ′[A(t)][Ck(t,t′) φk(t′)], (47)
− V −
The validity of these assumptions could be checked nu-
merically. Sincethishasbeendoneforequationsofsimi-
[T(µ+k2)+∂/∂t]Rk(t,t′)=δ(t t′)+ lartypeelsewhere[35],weomititinthepresentanalysis.
−
t
+4 dt [B(t,t )]r(t,t )[R (t,t) R (t ,t)] Second, it is well known that asymptotic solutions of
′′ ′′ ′′ ′′ k ′ k ′′ ′
Z0 V − such equations can be characterized by a few order pa-
2β [A(t)]R (t,t), (48) rameters [34,35,39–41]. They are defined as
0 ′ k ′
− V
q˜ = lim C (t,t), (56)
[T(µ+k2)+∂/∂t]φ (t)= k t k
k →∞
t qk = lim Ck(τ), (57)
4 dt′′ ′′[B(t,t′′)]r(t,t′′)[φk(t) φk(t′′)] τ→∞
Z0 V − q0,k = lim qkCˆk(λ), (58)
+2β0 ′[A(t)](Γk φk(t)), (49) λ→0
V − ϕ = lim φ (t). (59)
k k
t
→∞
t
(µ0+k2)φk(t)=2β0 dt′′ ′[A(t′′)]Rk(t,t′′), (50) The following k-integrated quantities will also be useful:
V
Z0
dk
q˜ q˜ = lim x(s,t)x(s,t) , (60)
(µ0+k2)Γk =1, (51) ≡Z 2π k t→∞h i
dk
q q = lim lim x(s,t)x(s,t+τ) , (61)
k
whereB(t,t)andA(t)aredefinedasB(t,t)= (x(s,t) ≡ 2π τ t h i
x(s,t))2 =′C(s,t;s,t)+C(s,t;s,t) 2C′(s,t;hs,t) an−d Z dk →∞ →∞
′ ′ ′ ′
ΓA((st,)s=). Nhi(oxt(es,tht)at−dxu0e(tso))t2rian=slaCti(osn,at;l−sin,tv)ar−ia2nφc(esw,ti;ths)r+e- q0 ≡Z 2πq0,k =λli→m0tl→im∞hx(s,t)x(s,λt)i, (62)
dk
spect to s both B(t,t′) and A(t) are s-independent. The ϕ≡ 2πϕk =tlimhx(s,t)x0(s)i. (63)
equations of motion for Ck(t,t′) and Rk(t,t′) are almost Z →∞
identical to those for the pure RHP model. Coupling to
thenativestateentersthroughthetermsproportionalto q˜ measures the size of the globule, q measures the per-
β . Again, for large selection temperature β 0 and sistent correlation in the TTI regime, q the asymptotic
0 0 0
→
one recovers the RHP model. correlation in the aging regime, and ϕ the overlap with
native state. Also, it is useful to define
b=2(q˜ q), b =2(q˜ q ), (64)
VII. EXTRACTING ORDER PARAMETERS 0 0
− −
1
a lim [x(s,t) x (s)]2 =q˜ 2ϕ+ . (65)
0
The equations of motion are coupled integro- ≡t→∞h − i − 2√µ0
differential equations with initial conditions given by
C (0,0),φ(0),and(weuseIto’sconvention)R(t+ǫ,t)
k → Third, we assume that the generalizedfluctuation dis-
1 as ǫ 0. To solve the equations analytically we have
→ sipation theorem is valid in the form
to consider several assumptions (which can be checked
by numerical solution).
x dCˆ (λ)
First, we make the (rather strong) standard assump- Rˆk(λ)= qk k , (66)
T dλ
tionsfromagingtheoryforspinglassesabouttheasymp-
totic behavior of the solutions: In the regime of time
translational invariance (TTI), x could in principle depend on k and Ck. However, re-
latedmodelshavebeenstudiedindetailandtheyexhibit
lim C (t+τ,t)=C (τ), (52) onestepreplicasymmetrybreakingwithak-independent
k k
t x. Thisonestepreplicasymmetrybreakingansatzinour
→∞
lim R (t+τ,t)=R (τ), (53)
k k dynamical study translates exactly to Eq. (66).
t
→∞
6
1
VIII. RELATING ORDER PARAMETERS b= , (75)
√µ˜+Σ
For t=t and t Eq. (47) gives
′
→∞
1 1 x 1 1
T(µ+k2)q˜ =T + 2 (b)(1 x)(q˜ q ) b0 = x√µ˜ + −x √µ˜+Σ, (76)
k ′ k k
TV − −
2
+TV′(b0)x(q˜k−q0,k)−2β0V′(a)(q˜k−φk). (67) b (b ) 1 1 µ 2
q˜= 0 + V′ 0 + − µ˜
With t = t′+τ and t′ and then τ Eq. (47) 2 4T2µ˜3/2 4√µ0 1− µµ˜0! ×
→ ∞ → ∞
becomes µ µ 2
0 0
2+ 1 , (77)
T(µ+k2)qk = 2( ′(b) x ′(b0))(q˜k qk) ×(cid:18) r µ˜ (cid:19)(cid:18) −r µ˜ (cid:19)
T V − V −
2
+ (b )x(q˜ q ) 2β (a)(q φ ). (68)
TV′ 0 k− 0,k − 0V′ k− k b0 ′(b0) 1 1
a= + V +
2 4T2µ˜3/2 4√µ0 1+ µ0 2 ×
Eq. (47) in the aging regime t =λt, first for t and µ˜
′
then λ 0, gives →∞ µ µ (cid:16) µqµ (cid:17)
→ 0 1+2 0 +2 0
× µ˜ µ˜ µ˜ µ˜
T(µ+k2)q0,k = T2V′(b0)(1−x)(q˜k−qk) (cid:20)r µ(cid:18) 2 r (cid:19)µ r
0
2 + 2+ , (78)
+ ′(b0)x(q˜k q0,k) 2β0 ′(a)(q0,k φk). (69) (cid:18)µ˜(cid:19) (cid:18) r µ˜ (cid:19)#
TV − − V −
Eqs. (49) and (50) result in two equations for ϕk, 2
µ˜=µ+ (a), (79)
′
2 TT0V
T(µ+k2)ϕ = (a)(Γ ϕ ) (70)
k ′ k k
T V −
0
2 and the combination of Eq. (75) and (72) gives
(µ0+k2)ϕk = ′(a)x(qk qo,k)
TT V −
0 0=rˆ(1) T2+b3 (b) . (80)
2 V′′
+ (a)(q˜ q ) (71)
TT V′ k − k (cid:2) (cid:3)
0
Furthermore,the overlapϕwiththenativestateisgiven
by
They are equivalent; one can chose to solve for the order
parameters workingwith either (70) or (71). This seems
1 1 µ
aratherremarkablecoincidence. We believe thatitorig- ϕ= − µ˜ . (81)
inates from the SUSY invariance of the original action 2√µ01+ µ0
µ˜
S. For example, a similar comment holds for equations
q
(47)and(48);they areequivalentinthe TTIregimeand
one can derive one from the other. The ‘conspiracy’ of All overlaporder parameters are positive. However, this
(49) and (50) not contradicting each other is very likely result is not obvious and has to be obtained after some
a similar phenomenon. Eq.(48) for λ=1 reduces to algebra.
These equations have two kinds of solutions. In one
4 (b)
Rˆk(1)(µ˜+k2+Σ)=−(q˜k−qk) VT′′2 rˆ(1), (72) kind, b = b0, so there is no glassiness (aging). For this
kind of solution, the parameter x is irrelevant. We call
such solutions “ergodic”. (While it will turn out that
where
some of them are not truly ergodic, in the sense of de-
dk scribingstateswheretheentireconfigurationspaceisvis-
rˆ(λ) Rˆ (λ) (73)
≡ 2π k itedwithBoltzmannprobabilities,theyviolateergodicity
Z inarathertrivialway,likeaferromagnetbelowtheCurie
temperature. We could call them “non-glassy”, but we
and Σ is defined by
prefer not to use a negative term.)
2
Σ=xT2 (V′(b)−V′(b0)). (74) For an ergodic solution, with b = b0, Σ = 0. Further-
more, rˆ(λ) = 0, so Eqn. (80) is trivially satisfied. One
then has to solve the four equations (75) and (77-79)for
Solving these equations for the orderparameters gives b, q˜, a and µ˜.
7
The stability of such a phase against glassinesscan be a= 1 + V′(√1µ˜) + 1 1
d(seeteerFmigin.e1d).uTsihnegret,hweeanstaulydsieisdwaempordeseelnwteitdhinnoPnaapteivreI- 2√µ˜ 4T2µ˜3/2 4√µ0 1+ µµ˜0 2 ×
state bias in its interactions (T0 = ∞) for finite µ. The µ0 µ0 (cid:16)µ qµ0 (cid:17)
boundaryoftheglassystateasafunctionofµhasaform 1+2 +2
× µ˜ µ˜ µ˜ µ˜
qualitatively like thatin the p-spinglass asa function of (cid:20)r (cid:18) r (cid:19) r
field [42,43]. In the present model, the presence of the µ 2 µ
0
+ 2+ (85)
native state enters the calculation solely through the re-
µ˜ µ˜
placement of µ by µ˜. Therefore, if a particular T and µ˜ (cid:18) (cid:19) (cid:18) r (cid:19)#
2
fall in the glassy regime (the region below the full and
µ˜ =µ+ (a) (86)
′
dashedlines)inFig.1,theergodicansatzhastobegiven TT0V
up.
They can be solved numerically: given µ, µ , T and T
0 0
The instability can occur in two ways, according to
onecanfindµ˜,whichinturndeterminesq˜,b=b (equiv-
0
whether µ˜ is bigger or smaller than the critical value µ˜ .
c alently q = q ), and ϕ. However, it is possible to gain
0
Aboveµ˜ ,thelineseparatingglassyfromergodicregions
c some analytic understanding in a few soluble limits.
is an Almeida-Thouless (AT) line; below it the stability
condition In this discussion we will concentrate on the limit of
small µ. As we noted in paper I, if we want to confine
T2+b3V′′(b)>0, (82) nNeemdoµnomNers4w/dit.hiTnhuasg,yforartaionlonragdpiuosly√mq˜er∝µµ−10/4.,Wwee
−
∝ →
will also take µ=µ to simplify the algebra a bit.
0
is violated. For µ˜ < µ˜ , there is no AT instability. The
c
transition is like that for the completely random het- The pair of equations (85) and (86) fully determine µ˜
eropolymer. To find such a transition, we have to solve as function of T and T0. For µ0 =µ they take the form
for a glassy phase, characterized in part by a value of 1 B˜2
the FDT-violation parameter x<1 and then find where a(µ˜)= +
in the parameter space x 1. In the region where the 2√µ˜ 8T2µ˜3/2(σ+µ˜−1/2)d2+1
→
x<1 solutionexists, the associatedergodic phase is un- 1 µ
+ 1+ (87)
stable and is replaced by the glassy one. 4√µ˜ µ˜
(cid:18) (cid:19)
In a glassy phase, aging is present: rˆ(1) = 0, so the B˜2
quantity in brackets in Eq. (80) has to vani6sh, i.e., the µ˜(a)=µ+ (88)
AT condition has to be satisfied as an equality, rather TT0(σ+a)d2+1
thananinequality. Thisso-calledmarginalstabilitycon-
Given µ˜, T, T one can find the overlap with the native
dition determines b as a function oftemperature. In this 0
state ϕ and the size of the polymer from (81).
case we have three more unknowns, Σ, b and x, making
0
a total of seven, and seven equations, (75-80), to solve
for them.
A. Random-globule state
We look for ergodic solutions first in the next section,
and we examine their stability. Then, in the following
section, we study glassy solutions (within the 1-step ag- It is immediately evident that when both the tem-
ing ansatz of section VII) and identify the regions in the perature T and the selection temperature T are large,
0
parameter space where they hold. µ˜ µ in (86), leading to a random-globule solution
≈
a = b = µ 1/2, q˜ = µ 1/2/2, ϕ = 0. What is not so
− −
obvious is that in the µ 0 limit a solution very close
→
tothis existsallthe waydownto verylowtemperatures,
IX. ERGODIC PHASES
even for small T . In this subsection we examine this
0
state in detail.
For ergodic phases, Eqns. (75-79) reduce to We look first for solutions of Eqs. (87) and (88) with
the ansatz α µ˜/µ fixed and µ 0. We call this the
≡ →
1 randomglobuleansatz,since,aswillbeshown,the poly-
b=b = (83)
0 √µ˜ mer does not have any fixed conformation(it is melted),
andontheaveragetheconformationsitadoptshavezero
q˜= 1 + V′(1/√µ˜) + 1 1− µµ˜ 2 overlap with the native state. (Strictly speaking, this is
2√µ˜ 4T2µ˜3/2 4√µ0 1− µµ˜0! × the only truly ergodic phase we find.) For a we get,
µ µ 2 1 3+ 1
2+ 0 1 0 (84) a= α + (µ(d−2)/2) , (89)
× µ˜ − µ˜ √µ 4√α O
(cid:18) r (cid:19)(cid:18) r (cid:19) (cid:20) (cid:21)
8
which, after inserting into (88), gives 3B˜2 d d−2
+1 µ 4 <1 (96)
4TT 2
B˜2 4√α d/2+1 0 (cid:18) (cid:19)
α 1+ µ(d 2)/4 . (90)
≈ TT0 − (cid:18)3+ α1(cid:19) for the existence of a random-globule-like state.
Some caution is in order. Working this out for finite
Eqn. (90) can be used to calculate α as a function of µ.
N, d = 3, and an average density of 1, we find that the
One can see easily that α 1 when µ 0. This shows
inequality (96) is violated below a temperature
→ →
that our ansatz is self-consistent in the limit of small µ.
Also, (84) and (81) become π 1/2 15B˜2
T = N 1/3. (97)
x −
1 α 1 6 8T0
ϕ= − + (α 1)2 (91) (cid:16) (cid:17)
2√µ 2 O −
(cid:20) (cid:21)
1 With the small power of N−1, one has to go to quite
q˜= [1+ (α 1)]. (92) large N to make this temperature very low. Thus our
2√µ O −
statement that the random-globule-like state exists for
all temperatures in the µ 0 limit may be of limited
→
The normalized overlap between the polymer confor- relevance for real 3-dimensional heteropolymers of the
mation and the native state is: length of typical proteins. Nevertheless, here we are just
considering this simple limit.
lim x(s,t)x (s) ϕ
t 0
cosθ = →∞h i = . (93) We now discuss the stability of this solution. In the
limt→∞hx(s,t)2ihx0(s)2i q˜(2√1µ) large-N limit,itislocallystableagainstspontaneousfor-
p q mationofanative-likestateatanyT andT0. However,it
¿From (91) and (92) we get cos(θ) (α 1) µ(d 2)/4. is unstable against glass formation at low temperatures:
−
∼ − ∼ Since it is identical with the random-globule solution of
Thus there is no overlapwith native state as µ 0.
→ the completely random heteropolymer problem, we can
Furthermore, to check that polymer does not freeze takeovertheresultfrompaperIthatitisunstablebelow
into some other conformation, we calculate the normal- a temperature T B˜, with the constant of proportion-
g
ized overlap between two configurations taken at very ality of order 1. T∝his glass temperature is independent
different times, of T . (In Fig. 1 this is the transition at µ˜ 0.) Thus,
0
→
wherever the system is in a random-globule-like state at
lim x(s,t)x(s,t+τ) q
cosθ′ = τ,t→∞h i = . (94) T > Tg, it will no longer equilibrate if the temperature
[limt→∞hx(s,t)2i]2 q˜ is lowered below Tg. Instead, it will become glassy and
its dynamics will show aging.
p
After rewriting
b 1
q/q˜=1 =1 , (95)
− 2q˜ − 2√µ˜q˜ B. Ergodic native state
and, using (92), we get cosθ = (α 1). Again, as
′
O − AtlowT andT,oneexpects that the polymershould
µ 0, cosθ 0. This confirms that the ansatz 0
′
→ → be very close to its native state, i.e., small a. Therefore
α = (1) and µ 0 leads to a melted random-globule-
O → we alsolookfor suchsolutions ofEqs.(87)and(88). We
like phase. This phase is identical to that found at high
willtrytosolveequations(87)and(88)inthelimitwhere
temperatures for the completely random heteropolymer
µ 0 and µ˜ stays finite. The limit µ 0 turns out not
in paper I. → →
to involve any subtleties when µ˜ is kept constant, so we
The validity of the present ansatz rests upon the fact will just set µ = 0 from the outset. Eqs. (87) and (88)
that we can solve Eq. (90). Clearly,for µ 0 a solution become
→
can always be found, namely α = 1. Since the physi-
cally relevant µ is N 4/d, we can always satisfy this 3 B˜2
−
∝ a(µ˜)= + (98)
equation, for any T0, in the limit N →∞. 4√µ˜ 8T2µ˜3/2(σ+µ˜−1/2)d2+1
We now address briefly the question of what happens B˜2
forfiniteN (andµ). OnecaneasilyseethatEq.(90)has µ˜(a)= (99)
two solutions when µ(d 2)/4/(TT ) is not too large (e.g. TT0(σ+a)d2+1
− 0
by plotting the left- and right-hand side as functions of
α). The solution close to 1 is lost when the slopes of the Theseequationscanbe solvedforµ˜ asfunctionofT and
left- and right-hand sides become roughly equal. Evalu- T . However, one has to keep in mind that µ 0 has
0
→
ating these slopes leads to the condition been taken. This implies that (81) and (84) become
9
1 1
minimum) describing the ergodic native phase. For ex-
ϕ , q˜ , (100)
≈ 2√µ ≈ 2√µ ample,thesolutionlabeledµ˜ inpanel(a)isofthis sort.
2
Theonewiththesmallervalueofµ˜ (e.g.,theonelabeled
and,inserting(100)into(93),thenormalizedoverlapbe- byµ˜ inpanel(a))is unstable. Itdescribesafreeenergy
1
tween native state and polymer conformations, becomes maximum between the minima at the random-globule
cosθ 1. Furthermore,because of its largeoverlapwith and ergodic native states. We will call such states “un-
≈
the native state, the polymer is essentially frozen. This stable stationary” (abbreviated US). (We have not done
can be seen by calculating the normalized overlap be- a static calculation to show this, but the situation here
tween two polymer conformations after a very long time is analogous to that in an ordinary ferromagnet below
interval, as in previous section. Inserting (100) into (94) the Curie temperature. There, one has three solutions
and (95) gives cosθ 1 µ/µ˜ 1. of the mean field equations, one with positive, one with
′
≈ − → negative, and one with zero magnetization. The middle
There is interesting behavpior associatedwith the limit
one,withzeromagnetization,isunstable). The USstate
µ 0 for very long polymers. When the polymer gets
→ has a lower overlap with the native conformation than
longer and longer (N ) a finite part of the chain is
→ ∞ theergodic-nativesolutiondoes,becauseithasasmaller
not in the native state conformation, since a stays con-
value of µ˜. As β is increased from below through βmin,
stant. The rest of the chain is in the native state, which 0 0
the native-state and US-state solutions appear together
can be seen from the fact that overlap with native state
and separate. For the temperatures of panels (a)-(c),
approaches 1. Thus, in the limit of a very long polymer,
they both exist for all β >βmin.
thefractionofchainnotinthenativestateconformation 0 0
becomes negligible: the recipe for biasing the coupling Panel (d) (at the lowest of the temperatures) shows a
constantsBss′ describedinchapterIIworksbestforlong morecomplexbehaviorwheredouble-minimumstructure
polymers. appears. We have found numerically that this happens
belowT 0.20. Herethebehavioraroundβmin isjustas
In the following we will proceed with the solution of ≈ 0
in the other cases, but we note that at this temperature
equations (98) and (99). Before continuing, it will be
β (µ˜,T) has a second local minima at a smaller value of
useful to compactify notation a bit. Making the change 0
µ˜. Thus there is a range βmin < β < βmax for which
ofvariablesXˆ =X/σforX =b, b , a, q, q˜;Yˆ =Yσ2 for 1 0 1
0 there arefour solutions. The rightmostone is stable and
Y = µ, µ˜; and Zˆ = Zσ(d 2)/4/B˜ for Z = T, T , we get
− 0 describes the ergodic-native phase, as before. Moving
equations of the same form, with X Xˆ, Y Yˆ and fromright to left, the solutions alternate between stabil-
Z Zˆ, but with σ = 1 and B˜ = 1. →Thus, wit→hout loss ityandinstability. Thusthesecondsolutionfromtheleft
of→generality, we can choose units with σ =1 and B˜ =1 represents a locally stable conformation. It is also corre-
(and remove the hats). From now on we do this. lated with the native state, since µ˜ is finite (though we
The working strategy for solving the equations is as always find µ˜ 1 in 3 dimensions). The remaining two
≪
follows. For fixed T, one can consider T0 as a function solutions (with ∂β0/∂µ˜ < 0) represent US states (local
of µ˜. This canbe easilydone by inserting the expression free energy maxima) between it and the random-globule
for a from Eq. (98) into (99), thus writing β =1/T as phase in one direction and the ergodic-native phase in
0 0
the other.
β (µ˜,T)=Tµ˜[1+a(µ˜,T)]d/2+1 (101)
0 PlottingβminagainstT,weobtainthestabilitybound-
0
ary indicated by the thick solid curve in Fig. 3. Within
The four panels of Fig. 2 shows the shape of β (µ˜,T) as
0 our present assumption of ergodicity, everywhere to the
a function of µ˜ for four different temperatures. We want
right of this line the ergodic-native phase is dynamically
ultimately to construct a phase diagram in the (β0,T) stable. One can invert the relation βmin(T), obtaining
0
plane. Therefore we have to specify T (one panel of the
a transition temperature T (β ), the maximum temper-
n 0
figure) and β and ask whether one or more solutions,
0 ature for which the ergodic native phase is dynamically
i.e., particular values of µ˜ which solve Eq. (101), exist.
stable. It is separated from the (also stable) random-
For example, in panel (a) in Fig. 2, a horizontal line at
globule phase by a barrier, the top of which is described
β > βmin intersects β (µ˜,T) curve at two places, in-
0 0 0 by the unstable solution.
dicating two solutions µ˜ = µ˜ ,µ˜ . To make the figures
1 2
In Fig. 3 we also indicate the region in the (β ,T)
morereadablewehaveshownsuchahorizontalline, ata 0
plane where the second locally-stable solution is found.
particularvaluesofβ , onlyin panel(a). If this horizon-
0
tal line is moved below βmin, it will never intersect the Thisregionhastheformofakindofsliverextendingout
0
toward large β at low temperatures.
β (µ˜,T) curve. Thus, we can see that for every T, there 0
0
is a value β0min(T) below which no solutions exist. So far we have not examined the stability of these so-
lutionsagainstglassiness. Asindicatedabove,wedothis
WeproceedwiththeanalysisofFig.2. Forsufficiently
with the help of Fig. 1: Stable solutions can not lie in
high temperatures (panels (a)-(c)) there are exactly two
solutions for all β > βmin. Of these, the one with the the range µ˜min < µ˜ < µ˜AT. In Fig. 2, these limits are
0 0
markedontheµ˜axes. Wethussee,forexample,inPanel
larger value of µ˜ is a stable solution (local free energy
10