Seshagiri, N.

English

Publications of the IAS Fellows

MECHANISM OF FOLDING AND DISULPHIDE I 
BOND FORMATION IN GLOBULAR PROTEINS: I 
A PROPOSAL FOR A UNIFIED THEORY I 
N" SESHAGIRI i 
National Information Centre, E-Wing, Pushpa Bhawan, Chiragdelhi - Madangir Road, New ~elhi -
110062, India I 
StnIcture of the Medium of Folding I 
Several hypotheses have so far been advocated in an effort to explain and simulate the 
folding of protein chains. On the assumption that these hypotheses are based bn partial 
sets of experimental results representing features of a possible unified theory bf protein 
folding, this paper attempts to provide a conceptual framework which may codtribute to 
the evolution of such a unification. I 
Perhaps one of the least understood mechanisms is that involving the interadion of the 
chain with the surrounding aqueous medium. The average diameter of the watet molecule 
of approximately 4 A indicates that the number of atoms in the water molecbles in the 
I 
region of importance to the folding will be comparable to the number of atoms in the pro-
tein molecule. Simulation models should therefore include water molecules indiyidually as 
far as possible, though statistical and thermodynamic analysis may guide such a simulation. 
Frank and Wen [1] and Frank [2] postulated the existence of flickering clusteh of water 
molecules which form hydrogen bonds in a co-operative manner. This idea was utilized 
by Nemethy and Scheraga [3,4] in characterizing the medium of protein foldingl However, 
the representation here is basically different in that we assume a continuum of ~ater mol-
ecules with a variety of distorted hydrogen bonds in such a manner that the continuum 
• responds to small local fluctuations of energy by varying the distortion in the bonds. 
A mathematical description of such a" continuum is given in the following: 
The total effective pair potential between two water molecules i and j wit positional 
co-ordinates (R;, Rj ) is given in the usual form, 
U(i,j) = ULJ(R;j)+UEL(i,j)+UHB(i,j), (1) 
where, ULJ(Rij) is the Lennard-Jones part (R < 2 A), UEL(i,j) is the dipole dipole part 
(R ::> 5 A), and UHB(i,j) is the hydrogen bond part (2 A",,; R",,; 5 A). 
As a regular hydrogen bond occurs only along the four directions pointing to the vertices 
of a regular tetrahedron, four unit vectors, i(La), i(Ha), j(L{3)' and j(H{3)' are i troduced in 
each molecule. Also, G(x) refers to an unnormalized Gaussian function equal to e p( - x2j202), 
where 0 2 is the variance. A bond will form whenever the direction of the O-H !bond of one 
molecule lies along the direction n = Rijl Rij and the direction O-L of the secohd molecule 
simultaneously lies along the direction (- n). From the point of view of *onte Carlo 
169 I 
I 
170 N. SESHAGIRI 
, 
simulations of the class considered by Ben Nairn [5], the configul'ation favouring H-bond 
formation for any pair (ex, fJ) may be taken to satisfy, , 
-a' < Rij-RH < a', I 
-a < i(A).n-l < a, 
-a < j(B).n+l < a. 
(2) 
I 
Here a = 2.85 A and a' is the rate at which the energy drops at 'the distance I Rij- RHI, 
where RH is the intermolecular distance at which maximum binding energy is obtained. 
(A, B) is either (Ha., L[3) or (La., H[3). A configuration of the pair (i, J) is hydrogen-bonded if 
and only if it satisfies conditions (2). 
For simplicity of simulation, the system of m-sequential hydrogen bonds may be assumed 
to favour the presence of the ,oH ... ,oH ... ,oH and related chain structures. Assuming 
a co-operative bonding in the m-sequence it is possible to derive the following condition 
for its feasibility utilizing the results of Pople [6, 7], Donohue [$], and Goel et al. [9]: 
-ma'/I(Gm) < f (Rijs-RHs) < ma'!l(Gm),l 
,=1 
m 
-ma !2(Gm) < L (ilAs)·ns-l) < ma!2(Gm), (3) 
s=l 
m 
-ma!2(Gm) < I (js(B,).ns+l) < ma!2(Gm), I 
s=l 
1 m 
G m = 1 + -u L ksm(l- cos <Psos). 
m s=1 
(4) 
Here 8s is the angle of bend between the respective ,oH and H ... ,0. ksm is the effective 
hydrogen bond bending force constant which depends on m, and the relative bending of the 
sth bond in the sequence relative to its neighbours taking into consideration non-additivity 
factors. Os = 1 if <Ps >- <Po and 0 if <Ps ~ <Po. Though the calculati~ns of Goel et al. [9] 
indicate that <Po "'" 25°, better results are obtained by assuming <Po <',15°. ksm may be more 
conveniently written as koksm, where k sm is a factor depending on t~e relative bending in 0 
the neighbourhood and ko (= 3.78 X 10-13 erg rad -2) is the bond behding force constant. 
I 
U is the minimum energy of an open dimer'!l(Gm) and.t;(Gm) are functions of Gm which 
can be approximately expressed as C1Gm and C2Gm respectively, where C1 and C2 are 
semi-empirical constants. The single H-bond case, which is a modification of the conditions 
(2), is derived by letting m = 1 and k sm = ko. 
Mixed Hydrophobic-Hydrophilic Cavity Effect 
The possibility that the introduction of the protein chain in a water medium decreases 
the density in the neighbourhood and creates conditions favouring the formation of cavities 
is examined as a prelude to the quantification of hydrophobic and hydrophilic behaviour 
of side chains. 
---------
....... ; ... '->. -
MECHANISM OF FOLDING AND DISULPHIDE BOND FORMATION IN GLOBULAR PROTEI S 171 
Based on the two structure model, the total differential of the volume V, with crnstant 
T and P, can be expressed as, _ 
I 
Here, s, I, and h are the number of solute molecules, average number of L-cules df water 
and average number of H-cules of water respectively. I 
Imposing the condition dl+dh = 0, eqn. (5) can be rewritten as I 
where 
(5) 
(6) 
- 'I
, dV= V.ds+(VL-VH)dl. I (7) 
As T, P, and the number of water molec~les w (s -< w) are constants and /.is not tn inde-
, pendent variable, I 
• 
dV = Vs ds+ (VL-VH) (:!t ds. (8) 
The derivative is taken along the equilibrium line for the reaction L ~ H. With the ~ddition 
of ds molecules to the system keeping I and h fixed, the measurable change in v6lume is 
I 
given by the first term of eqn. (8). Releasing the constraint imposed by fixing I and? relaxes 
the system to a new equilibrium position with the measurable change in volume p'ven by 
the second term of eqn. (8). Similar expression can be derived for enthalpy which gives the 
degreee of hydrogen bonding. I 
The change in volume in eqn. (8) propagates a pressure on the continuum whose magni-
tude can be expressed through 
(OV) oVs oVL oVH (01) oP T,w = s oP +1 oP +h oP +(VL-VH) oP w· (9) 
Here it can be shown that along the equilibrium line for L ~ H 
(a/) -V(VL-VH)2XLXH'YJ I' (10)-(VL-VH) oP T,w kT 
o,where XL and xHare the respective mole fractions and 'fJ is the volume density. The fncreased 
pressure is partly absorbed by bent m-sequences in the continuum in accord~nce with 
eqns. (3) and (4). I 
A measure of the hydrophobic driving force was made by Tilnford [10], Brapdts [11], 
and Epstein [12]. The hydrophobic residues have a tendency proportional to t~is hydro-
phobicity factor HfA' for creating cavities in the continuum by virtue of their lack of 
affinity for water molecules. The work required to form a cavity in ahydrophobif environ-
ment can be quantified by known ther~odynamic methods. The probability ~ensity of 
observing any specific configuration RW is given in the usual form, I 
I 
I 
I 
(11) 
.•. H.I.. _,I" .1. ,I II" I 'I j 
172 N. SESHAGIRI 
, 
The probability of finding the centres of all particles in a regior S is 
P(S) = f ... f dRwP(Rw). (12) 
s 
, 
If all the centres are to be excluded from a volume V(Ro, r) formed by a spherical cavity of 
radius r and centre at a fixed position Ro, 
pc.iRo, r) = f ... f dRWP(RlV) = exp[ -(3W~Ro, r)], (13) 
V-VCRo-r) , 
where W(Ro, r) is the work requiredior the creation of the cavity. ',Conversely, given W the 
radius r of the cavity can be obtained from eqn. (13). 
In the folding process, if a tendency for cavity formation exis~s because of decrease in 
density of hydrophilic forces nearby, the volume of the cavity 'may be accentuated by 
W oc HfA for the side chain of a given amino acid A. The volume Gontribution of this W to " 
the overall cavity can be obtained by eqn. (13). 
The foregoing derivations point to a reduction in the density of the medium around the 
protein chain as well as in the interior of the folds as compared to the bulk medium. As the" 
number of atoms in the chain and those of the water molecules in the domain of folding 
are comparable, the volume of cavity will be appreciably high. :As the hydrophilic side 
chains attract water molecules and the hydrophobic ones do not, the structure of the con-
tinuum in the interior, as the chain folds, is likely to be in the form ?f a molecular grid with 
the continuum spanning from one hydrophilic part to another, the, connectivity depending 
upon the spatial distribution of the various hydrophilic and hydrophobic parts. At any 
stage of the folding we can determine the approximate cavity-to-oontinuum volume ratio 
in the following manner. . 
The increase of volume in the presence of the neighbourhood of ~ folding chain is deter-
mined by a non-linear interpolation between that for the linear ch~in and the fully folded 
globular form. For the linear chain the net increase in volume due to the addition of solute 
particles alone is given by integrating eqn. (8) over the entire vol~me of the linear chain. 
This is given as 
(14) 
I 
computed over the entire volume Vc of the linear chain. It is further noted that the hydro-
phobic behaviour of the side chains will create a tendency for cavity fqrmation enhancing the 
volume increase of eqn. (14). This tendency can be taken to be proportional to Pcav(Ro, r) of 
eqn. (13) integrated and normalized over the entire volume Vc' The ~ork W(Ro, r) is taken 
to be a function COHhA computed for each amino-acid residue A = 1,2, ... , N for creating 
a cavity of radius r = 1 and represented as W(RA' 1). Here Co is an pndetermined constant 
which is evaluated from trial simulations using known protein structures of small chain 
lengths. The net increase of volume is given by 
VI = C1 f [(1 +exp( -(3CoHhA)(VL - V H) (~~)J ds. (15) 
(V,. A) 
...... =-=,_ ...... _.=-me 
MECHANISM OF FOLDING AND DISULPHIDE BOND FORMATION IN GLOBULAR PROTEIN 173 
I 
Here the integration is carried out over all the solute molecules r~presenting the!· chain 
taking care to see that HhA refers to different A, corresponding to different sets of/solute 
molecules. C 1 is a proportionality constant which is determined similar to Co. I 
The fundamental notion behind the continuum is that the average density of the hy~rogen­
bonded water molecules is the same whether it is the bulk water medium or the vicibity of 
the chain. The increase in volume VI will, therefore, be possible only if we assume thb exist-
ence of cavities in the interior and the exterior neighbourhood of the chain. In view bf this, 
VI itselfwiIl be the volume of the cavity in the neighbourhood of the chain. I 
The volume of the cavity V(f) in the neighbourhood of a folding chain in the interrediate 
stage is approximately interpolated as i 
N(J) I Vi = C2 NI VI. (16) 
'- Here NI and N(fJ are the minimal number of water molecules in the vicinity of t~e chain 
that is of relevance to folding for the linear chain and the intermediate folded cil ain re-
spectively. . 
; The distribution of the cavity volume within the domain of interest to folding ~s taken 
as a non-linear function increasin! towards the instantaneous centre of mass of the inter-
mediate folded chain. For purposes of the present simulation model, we take I 
C5 Cn 
V(r) = C4 + (r+C3) + (r+C3)2 ' (17) 
f f f V(r) dV = V(f), (18) 
v I 
where V is the folding domain and C3 to Cn are constants. The folding domain han itsel 
be divided into an interior domain and an exterior domain for purposes of si±ulation. 
The int.erior domain is taken as the three-dimensional envelope of the straight linles drawn 
from each residue to the others. For more accurate simulations, it is necessary to define 
• a .kernel of the backbone. The kernel for the backbone for a linear chain is apprrximately 
the straight line joining the NH2 and COOH terminii and that for the fully foldtjd protein 
is the centre of mass. An interpolation between the two can be made for intermedihte stages 
•. using well-known geometrical methods for obtaining the intermediate kernels. T1~ integra-
tion in eqn. (18) is worked with respect to the nearest point on the kernel to the P'l'int under 
consideration. . ! 
The determination of the structure of the grid along with path connectivities! and path 
cross-sections require a knowledge of the manner in which polar residues will f affect the 
grid paths. I 
The simplest approximation assumes that each hydrophilic side chain can beJrepresent-
ed in the domain of the intermediate folding stage by a set of positive and negat'lve charges 
bearing the same polarity index as given by Epstein [12] and drawing the flux liIljes by well-
known electrostatic methods. For more accurate computations, methods ba~ed on the 
well-known Hellmann-Feynman theorem can be utilized. The continuum is acca1mmodated 
around the kernels of the flux field joining the hydrophilic side chains. The crosJ-section of 
I 
the path is made proportional to the flux density along the kernels. A better approximation 
I 
, 
174 N. SESHAGIRI 
recognizes the ionic, dipole factors present in the water medium and draws upon certain 
techniques normally employed in the thermodynamics of ionic hydration as well as calcula-
tions of the above type. 
With the determination of path connectivities and path cross-sections in the above form, 
we can distribute the bonded water molecules along the grid paths keeping the deviation 
of cavity volume distribution from that given eqns. (17) and (18) as minimum, i.e. 
f Sf [V(R) - V(R)J2 dR ->- minimum (19) 
v 
subject to Sf f VCR) dR = f Sf VCR) dR = V(J). (20) 
v v 
With the grid structure postulated above it is possible to associate various means of 
energy transfer like electron migration in conductivity bands, transfer by proton, hydride 
ion or hydrogen radical migration, transport by orientational defects, etc. A survey of~ 
possible mechanisms of energy transfer has been made by Steinberg [13] and von Hippel 
[14]. In this background we postulate the existence of a vector current (energy carrier) 
through the grid paths. 
We define a current path between two fixed-charge locations on side chains as paths 
connecting them along the grid path with constant scalar currents, each scalar represent-
ing a distinct mode of energy transport. Along a grid path between two nodes of the grid 
two or more current paths may coalesce in the continuum giving ~ net current along the 
segment. Without going into the proof, it is stated here that the current along a grid path 
is approximately a superposition of the currents along the (constant) current paths. From 
this the net current at any point in the grid can be determined. 
The Grid-path Simulation of Folding 
A broad outline of the simulation is presented below: 
(1) As the tendency of folding is such that the cavity volume in the domain decreases 
from the linear chain to the globular chain, the folding patfuway is such that this 
decreasing trend is maintained. Further, the polar residues are the starting or ter-
minal points for the currents both to the interior and exterior', of the domain. 
(2) Apart from the grid-path concept proposed here, all other ~spects concerning the ~ 
steric forces and constraints are considered using established me~hods. 
(3) A set of special folding tendency functions have been defined Jor oc-helices, ,B-bends, 
etc., which are included in the simulation as indicators of sensitivity of folding. 
(4) In the present routine, certain approximate gross representation of the features of 
vector currents with orientational features found between the cysteines acting through 
the grid-path medium have been assumed. This simulates the tendency for disulphide 
bond formation. 
(5) A feedback mechanism based on the following steps has bee~ incorporated: 
(a) The grid-path actuated mechanism mentioned in (1) and,(4) above introduces 
mechanical forces of different magnitudes with a tendency to place different side 
chains towards or away from the instantaneous grid kernel. 
- -- - -------- ------ ----------------- - - -----
I 
MECHANISM OF FOLOING AND DISULPHIDE BOND FORMATION IN GLOBULAR PROTEIJ 175 
(b) The mechanical forces contribute to the folding of the chain in addition t the 
forces mentioned in (2) and (3) above. I 
(c) The folding reduces the cavity volume as well as squeeze out a determinate n*mber 
of water molecules out of the domain with associated redistribution of the tavity 
volume. I 
(d) The grid paths reorient themselves to accommodate the new situation described 
in (c). Also, the vector currents over the current path are altered. 
(e) Step (a) is initiated again to begin the next iteration. 
Conclusion 
Concepts like water-molecular grid path and vector current have been advocated in the 
proposal for evolving a theory of protein folding for unifying the thermOdyna9ic and 
'nucleation pathway dependent theories. The simulation experiments, which are at present 
in a preliminary stage, have been carried out with (}2-microglobulin and BPTI with a boder-
ate degree of success. Until the simulation is carried out with bigger and more comple~ chains 
~using more accurate derivations, it is too early to compare the grid-path theory withlothers. 
Such an elaborate simulation package is currently under development. I 
I 
Summary 
Introducing the concept of "water grid pathways", a theory is advocated for the mechanism 
offolding and the formation of disulphide bonds in the tertiary structure of globJlar pro-teins~ The proposal is intended to give a framework for unifying the thermodynabic and 
kinetic approaches. 
References 
1. H. S. FRANK and W. Y. WEN, Disc. Faraday Soc. 24, 133 (1957). 
2. H. S. FRANK, Proc. Roy. Soc. Land. A, 247, 481 (1958) . 
• 3. G. NEMETHY and H. A. SCHERAGA, J. Chem. Phys. 36, 3382, 3401 (1962). 
4. G. NEMETHY and H. A. SCHERAGA, J. Phys. Chem. 66,1773 (1962). I 
5. A. BEN-NAIM, J. Chem. Phys. 54, 3382 (1971). 
7. A. POPLE, Proc. Roy. Soc. A, 205, 163 (1951). 
i 6. A. POPLE, Proc. Roy. Soc. A, 202, 323 (1950). I 
8. J. DONOHUE in Structural Chemistry and Molecular Biology (A. RICH and N. DAVIDSON, eds.), Freeman, 
San Franscisco, 1968. 
9. A. GaEL, A. S. W. MURTHY, and C. N. R. RAO, J. Chem. Soc. Land. A, 1971, 190 (1971). 
10. C. TAN FORD, Am. Chem. Soc. 84, 4240 (1962). 
11. J. BRANDTS, Biological Macromolecules, Marcel Dekker, New York, 1965. 
12. C. J. EpSTEIN, Nature 215, 355 (1967). 
13. I. Z. STEINBERG, Ann. Rev. Microbiology 747, 83 (1971). 
14. A. VON HIPPEL, J. Chem. Phys. 54, 145 (1971). 
.1. I· III 
l~: ____________ ~~ __ ~ ____ __ i: 


Mechanism of folding and disulphide bond formation in globular proteins: a proposal for a United theory

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Mechanism of folding and disulphide bond formation in globular proteins: a proposal for a United theory

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