# Protein photo-folding and quantum folding theory.

**ABSTRACT** The rates of protein folding with photon absorption or emission and the cross section of photon -protein inelastic scattering are calculated from quantum folding theory by use of a field-theoretical method. All protein photo-folding processes are compared with common protein folding without the interaction of photons (non-radiative folding). It is demonstrated that there exists a common factor (thermo-averaged overlap integral of the vibration wave function, TAOI) for protein folding and protein photo-folding. Based on this finding it is predicted that (i) the stimulated photo-folding rates and the photon-protein resonance Raman scattering sections show the same temperature dependence as protein folding; (ii) the spectral line of the electronic transition is broadened to a band that includes an abundant vibration spectrum without and with conformational transitions, and the width of each vibration spectral line is largely reduced. The particular form of the folding rate-temperature relation and the abundant spectral structure imply the existence of quantum tunneling between protein conformations in folding and photo-folding that demonstrates the quantum nature of the motion of the conformational-electronic system.

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**ABSTRACT:**Two classes of lambda(6-85) mutants (those richer in alanine, and those richer in glycine) have very similar slopes in an Arrhenius plot of the unfolding rates but very different temperature dependencies of the folding rates. Temperature-dependent interactions (e.g., hydrophobicity) play a large role in the initial stages of folding but not in the initial stages of unfolding of lambda(6-85). Placement of the transition state in terms of its surface exposure and entropy shows that at least two reaction coordinates are required to describe folding of all mutants over the full temperature range. The unusual Arrhenius plots of the very fastest mutant provide an additional kinetic signature for downhill folding.Biochemistry 11/2004; 43(41):13018-25. · 3.19 Impact Factor - SourceAvailable from: Steven John Hayward[Show abstract] [Hide abstract]

**ABSTRACT:**The biological function of proteins is associated with a variety of motions, ranging from global domain motion to local motion of side chain. We propose a method, dihedral transition analysis (DTA), to identify significant dihedral angle changes between two distinct protein conformations and for characterization of the effect of these transitions on both local and global conformation. Applying DTA to a comprehensive and non-redundant dataset of 459 high-resolution pairs of protein structures, we found that a dihedral transition occurs in 82% of proteins. Multiple dihedral transitions are shown to occur cooperatively along the sequence, which allows us to separate a polypeptide chain into fragments with and without transitions, namely transition fragments (TFs) and stable fragments (SFs), respectively. By characterizing the magnitude of TF conformational change and the effect of the transition on the neighboring fragments, flap and hinge motions are identified as typical motions. DTA is also useful to detect protein motions, subtle in RMSD but significant in terms of dihedral angle changes, such as the peptide-plane flip, the side-chain flip and path-preserving motions. We conclude that DTA is a useful tool to extract potential functional motions, some of which might have been missed using conventional methods for protein motion analysis. http://dynamics.iam.u-tokyo.ac.jp/DTA/Bioinformatics 02/2009; 25(5):628-35. · 4.62 Impact Factor - SourceAvailable from: Alan Fersht
##### Article: Direct observation of ultrafast folding and denatured state dynamics in single protein molecules.

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**ABSTRACT:**Single-molecule fluorescence resonance energy transfer (smFRET) experiments are extremely useful in studying protein folding but are generally limited to time scales of greater than approximately 100 micros and distances greater than approximately 2 nm. We used single-molecule fluorescence quenching by photoinduced electron transfer, detecting short-range events, in combination with fluorescence correlation spectroscopy (PET-FCS) to investigate folding dynamics of the small binding domain BBL with nanosecond time resolution. The kinetics of folding appeared as a 10-micros decay in the autocorrelation function, resulting from stochastic fluctuations between denatured and native conformations of individual molecules. The observed rate constants were probe independent and in excellent agreement with values derived from conventional temperature-jump (T-jump) measurements. A submicrosecond relaxation was detected in PET-FCS data that reported on the kinetics of intrachain contact formation within the thermally denatured state. We engineered a mutant of BBL that was denatured under the reaction conditions that favored folding of the parent wild type ("D(phys)"). D(phys) had the same kinetic signature as the thermally denatured state and revealed segmental diffusion with a time constant of intrachain contact formation of 500 ns. This time constant was more than 10 times faster than folding and in the range estimated to be the "speed limit" of folding. D(phys) exhibited significant deviations from a random coil. The solvent viscosity and temperature dependence of intrachain diffusion showed that chain motions were slaved by the presence of intramolecular interactions. PET-FCS in combination with protein engineering is a powerful approach to study the early events and mechanism of ultrafast protein folding.Proceedings of the National Academy of Sciences 11/2009; 106(44):18569-74. · 9.81 Impact Factor

Page 1

SCIENCE CHINA

Life Sciences

© The Author(s) 2012. This article is published with open access at Springerlink.com life.scichina.com www.springer.com/scp

email: lolfcm@mail.imu.edu.cn

• RESEARCH

PAPER •

June 2012 Vol.55 No.6: 533–541

doi: 10.1007/s11427-012-4316-9

Protein photo-folding and quantum folding theory

LUO LiaoFu

Faculty of Physical Science and Technology, Inner Mongolia University, Hohhot 010021, China

Received December 12, 2011; accepted March 28, 2012

The rates of protein folding with photon absorption or emission and the cross section of photon -protein inelastic scattering are

calculated from quantum folding theory by use of a field-theoretical method. All protein photo-folding processes are compared

with common protein folding without the interaction of photons (non-radiative folding). It is demonstrated that there exists a

common factor (thermo-averaged overlap integral of the vibration wave function, TAOI) for protein folding and protein pho-

to-folding. Based on this finding it is predicted that (i) the stimulated photo-folding rates and the photon-protein resonance

Raman scattering sections show the same temperature dependence as protein folding; (ii) the spectral line of the electronic

transition is broadened to a band that includes an abundant vibration spectrum without and with conformational transitions, and

the width of each vibration spectral line is largely reduced. The particular form of the folding rate––temperature relation and

the abundant spectral structure imply the existence of quantum tunneling between protein conformations in folding and pho-

to-folding that demonstrates the quantum nature of the motion of the conformational-electronic system.

protein folding dynamics, photo-folding, conformational change, quantum transition

Citation:

Luo L F. Protein photo-folding and quantum folding theory. Sci China Life Sci, 2012, 55: 533–541, doi: 10.1007/s11427-012-4316-9

Proteins are huge microscopic systems composed of several

thousands of atoms. In principle, proteins should obey

quantum laws. Recently, we proposed a protein quantum

folding theory [1–3]. Although bioinformatics studies, such

as the prediction of protein structure and function from mo-

lecular sequences, have achieved significant successes, the

dynamic problem associated with protein folding remains

unresolved. The proposed quantum folding theory empha-

sizes the concept of torsional cooperative transitions. The

importance of a torsion state can be examined as the fol-

lowing: a multi-atom system, the conformation of a protein

is fully determined by bond lengths, bond angles and torsion

angles (dihedral angles). Torsion angles are the most easily

changed of these three physical features, even at room tem-

perature, and are usually assumed to be the main variables

of a protein conformation. Simultaneously, the torsion po-

tential generally has several minima, with the transition

between minima responsible for conformational changes.

All torsion modes between contact residues are taken into

account in the proposed quantum folding theory. These

modes are assumed to participate in the quantum transition

cooperatively. In fact, the Bose condensation of strongly

excited longitudinal electric modes represents an example

of cooperativeness in living systems and was proposed in

the 1970s [4,5]. The fold cooperativeness of a protein was

also demonstrated in earlier literature [6]. These publica-

tions explained the possible existence of the cooperative-

ness in protein folding or in living systems from the point of

non-linear dynamics and thermodynamics. More recently,

contact order has been introduced as an important parameter

for understanding and calculating the folding rate [7].

Meanwhile, the dihedral transition was observed more di-

rectly in the statistical analysis of protein conformational

changes [8]. The cooperative dihedral transitions were

found to occur in most (~82%) polypeptide chains. Based

on the above considerations we formulated a quantum the-

Page 2

534 Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

ory on non-radiative protein folding. The point of folding as

a quantum transition can solve the folding speed problem

(Levinthal’s paradox)–how proteins can fold so fast when

they can sample so many possible configurations [9]. In

particular, the proposed theory can successfully interpret the

non-Arrhenius behavior of the temperature dependence of

folding rates [3].

To explore the fundamental physics behind the folding

more deeply and to clarify the quantum nature of the folding

mechanism more clearly we shall study the protein pho-

to-folding processes, namely, the photon emission or ab-

sorption in protein folding and the inelastic scattering of

photons on proteins (photon-protein resonance Raman scat-

tering). Although fluorescent proteins have been extensively

used as biological markers to observe gene and protein ex-

pression [10] and the fluorescence technique has been de-

veloped to examine protein folding and protein-protein in-

teraction dynamics [11], quantitative theory on fluorescent

transitions and its relation to protein folding remains unclear.

The reason for this absence of information may be the com-

plexity of the fluorescence mechanism, for example, many

active proteins need a cofactor or other small molecules to

fluoresce. An additional reason may be attributed to the

prevalent “too-classical” understanding of protein folding

and therefore the lack of a theoretical method to treat the

problem. However, the newly proposed quantum folding

theory affords a sound basis for discussing and studying

these problems. In fact, photon emission or absorption in

protein folding and the inelastic scattering of photons on

proteins, as an electromagnetic process, can be accurately

described by quantum electrodynamics. Because the elec-

tronic transition emitting or absorbing photon is coupled to

the conformational change of a protein; the torsion transi-

tion in a polypeptide chain plays an important role in deter-

mining the photon emission/absorption rates or cross sec-

tions. We shall make the first-principle-calculation on the

rates and cross sections of these photo-folding processes

based on quantum electrodynamics. The quantitative results

will provide several checkpoints on the quantum folding

theory. The experimental tests of these theoretical predic-

tions will provide refined evidence on the quantum nature

of protein folding and photo-folding.

1 Theoretical method: deduction of protein

photo-folding from quantum folding theory

1.1 Hamiltonian for protein folding and photo-folding

A protein is regarded as a conformation (torsion coordinate

{ }

j

mainly)––electron system. Protein folding is de-

scribed by the Hamiltonian

12

,( ,

,).HHx

In

adiabatic approximation the wave function of the system

can be expressed as

( , )

( ) ( , ),

Mxx

(1)

and these two factors satisfy:

2( ,

,)( ,

x

) ( )( ,

x

),

a

Hx

(2)

1

, ( )

( )

( ),

knkn kn

HE

(3)

where denotes the electronic state and (k, n) refer to the

conformation- and vibration-states, respectively. The Ham-

and the torsion potential term. The torsion potential gener-

ally has several minima with respect to each i and near

each minimum the potential can be approximately ex-

pressed by a potential of a harmonic oscillator. Any small

asymmetry in the potential has been shown to cause a strong

localization of the wave functions [1]. The localized con-

formational state is labeled by the quantum number k. Since

the adiabatic wave function is not a rigorous eigenstate of

the Hamiltonian H1+H2, there exists a transition between the

adiabatic states that result from the off–diagonal elements of

H1+H2. The transition describes the non-radiative protein

folding [1].

Since H2 contains the electronic kinetic energy term,

from gauge invariance of Hamiltonian H1+H2 we obtain the

electromagnetic interaction:

iltonian

1

,H

includes the kinetic energy term

22

2

j

2

j

I

(Ij is the inertial moment of the jth mode)

(EM)

1

H

(EM)

2 EM

,HH

(4)

(EM)

1

0

()

(),

2

e

Hcci

mc

V

k

k

kk

k

(5)

2

(EM)

2

2

0

'

(

4

),

e

Hc c

k

mc V

c c

k

c c

k

c c

k

k

k

k

k

kk

k

k

kk

(6)

where m is the electron mass, c

tion and production operators, respectively, for a photon

with the wave vector k, frequency kand polarization

,

εk

and V0 denotes the normalization volume. From the

perturbation H1

we calculate three types of protein photo-folding processes:

(i) the stimulated single photon emission and absorption

accompanying protein folding; (ii) the spontaneous photon

emission in protein folding; and (iii) the photon-protein

resonance Raman scattering. All calculations are carried out

by the quantum electrodynamics method. To simplify the

k and c

k are annihila-

(EM) to second order and H2

(EM) to first order

Page 3

Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

535

notation, the calculations are made in units of

and only in the final results the Planck constant ħ and the

velocity of light c are written explicitly.

1,c

1.2 Stimulated photon emission and absorption in

protein folding

We initially discuss single photon absorption. Set

,ikn

k

where

kn

)

and

k the photon number of the wave vector k and

polarization

.

k

For the multi-torsion case,

...,)

N

and

( )

kn

is the product of the functions of the

single argument. Likewise, set

cess is depicted by the reaction equation:

k photons + protein in (knα) →

(

1

k

) photons + protein in (k′n′′).

By using eq. (5) we obtain

~( , )

( )

( ,x

kn kn

Mx

12

(,,

,1

fk n

k

where

~ ( , )

( )

( , ).

x

k n

k n

k nMx

The above pro-

(EM)

1;

0

0

2

( )

( )d ,

2

k n

k n

P

kn

kn

e

m

f Hi

V

e

mV

P

k

k

k

k

k

k

(7)

where

;

00

( , )(

)( , )d d

x

( )

( )d

( ,

x

)()( ,

x

)d

( )

( )d

kn

kn

k n

P

kn k n

kn

kn

kn

MxiMx

ix

P

(8)

P

is the matrix element of the electron momentum. In

the above deduction of eq. (8), the Condon approximation,

namely, the matrix element

( , )(x

)( ,x)d ,ix

which does not depend on , has been used. The overlap

tion can be calculated under the harmonic approximation of

the torsion potential [1]. Note that because

wave function

( )

kn

and

to each other and the overlap integral always exists even for

.

kk

After taking into consideration the thermal average

over the initial vibration states and the summation over the

final vibration states we obtain the absorption rate:

integral

( )( )d

k n

kn

of the vibration wave func-

,

the

( )

k n

are not orthogonal

2

2

2

0

,

aV

e

WI

m V

P

k

k

k

(9)

2

{}

( )

( )d

( , )B n T,

j

V k n

kn Vj

npj

II

(10)

/2

1

exp{

(2 1)}(2(1))

j

j

p

j

n

Vjjjpjjj

j

n

IQnJQ n n

(11)

with

1

1

( 1),

j

j

B

ne

k T

(12)

2

( ) /2 ,

jjjj

QI

(13)

.

j

j

j

E

p

(14)

B(n,T) represents the Boltzmann factor for the thermal

average, j (′j) is the frequency parameter of the jth torsion

harmonic potential in the initial (final) state, Ej is the en-

ergy gap between the initial and final states (the minimum

of the jth initial potential minus that of the jth final) and j

is the angular displacement of the torsion potential (the po-

sition difference between two minima of the jth torsion po-

tential). pj represents the net change in the vibration quan-

tum number for the torsion oscillator mode j, which satisfies

the constraint:

j

j

pp

(15)

in the last summation of eq. (10). IV is the Thermo-Averaged

Overlap Integral (TAOI). By use of the asymptotic formula

for the Bessel function [12]:

1/2

2

( )(2π )exp(/2 )

z

p

e J zzpz

for z >> 1 (16)

Vj

I can be simplified. Finally we obtain

1

2

2

2

1

2π

(

)

= expexp

2 2()

Vj

Bj

GG

IZ

k TZ

－

(17)

with

2

j

2

(),

B

jj

k T

ZI

(18)

ln ,

j

B

j

j

GE k T

(19)

.

j

j

EE

(20)

is the average of the initial torsion frequencies

the oscillator mode j. In eq. (17) the energy gap E has been

replaced by the free energy decrease G to take the torsion

j

over

Page 4

536 Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

frequency difference

(or its average ),

jj

into account [1]. IV is a func-

(or its average ),

tion of

j

j

2

(

Note that the simplified expression (17) is obtained when

zj>>1. For photo-folding with a conformational change,

,kk

the condition is always fulfilled. The single photon

absorption cross section is obtained readily from (9):

)

j

(or its average

2

()

) and

j

E

(or its sum E).

2

2

2

π

( ,

,

, ).

aV

e

IE

cm

P

k

k

(21)

The corresponding absorption rate is denoted by Wa. By

comparison with the non-radiative folding rate [1]:

2

2π

( , , , ),

pfpf pfpf

fE V

I IkE

(22)

where IV is defined by eq. (10) and has the same simplified

expression as eq. (17), and the matrix element of the elec-

tronic wave function IE is given by

2

2

44

2

j

2

0

, , ,

44

j

Ejj

jj

j

a

I

A

IalAa

I

(23)

where I0 is the average inertial moment and lj represents the

magnetic quantum number of the electronic wave function

( , ).x

We obtain the ratio of rates:

2

0

2

2

22

( ,

,

,

,

,

pf

)

E

/(

m

)

2,

( , )

aV

pfpfpf

fV

W

k

I

A

IE

e

c

F

I

P

k

k

(24)

where

0

c

F

V

k is the incident photon flux in the pho-

to-folding process. Setting

15

~ 2π 10 ,

k

37

0~10,I

2

5

P

~10 ,

mc

4

~10 10A

(all in CGS units)

and

( ,

,

, )( , , , )

pf pfpf pf

VV

IEIE

leads to

2326

(10 10) .

F

a

f

W

k

Consequently, when the

photon flux is large enough, F > 1023 cm2 s1, the single

photon absorption rate is comparable with the protein fold-

ing rate.

The double- and multi-photon absorption rates or cross

sections can be calculated by the same method with the se-

cond and the higher order perturbations. As a general rule,

the absorption rates contain the TAOI factor, IV.

For a single-photon stimulated emission

k photons+protein in (kn) →

( 1)

k

photons + protein in (k′n′′).

Set

,i kn

k

calculating the matrix element

gle-photon stimulated emission cross section is deduced as

and

,+1fk n

k

. Through

(EM)

1

i Hf

the sin-

2

2

2

(

1)

π

(1).

eV

e

I

cm

P

k

k

k

k

k

(25)

The double- and multi-photon stimulated emission can be

calculated in the same way through the second and the

higher order perturbations. All results contain the TAOI

factor IV.

1.3 Spontaneous emission in protein folding and the

spectrum structure of photo-folding

Following the same perturbation approach and setting the

initial photon number

0

k

the stimulated emission, one obtains the single-photon

spontaneous emission rate. The rate of the quantum transi-

tion from a given initial state i

in the above deduction of

kn

to the definite

final state of

k photons

,f k n

k

is

2

(EM)

1

2π (

).

fi k n

kn

WEE f Hi

k

(26)

In the spontaneous emission case, the frequency of the

emitted photon is not given a priori because no stimulating

electromagnetic field of given frequency exists. Adopting a

continuous representation of the electromagnetic field ex-

pansion and replacing the sum over the photon final states

k

by

3

(2π)

width:

0 V

3

dk

we obtain the emission rate or partial

2

2

22

, .

2

knk n

eV

EE

e

I

c c m

P

k

k

k

(27)

The numerical estimate gives

8

~10

eVI

as

2

155

P

~ 2π 10 ,

~10

k

mc

(all in CGS

units).

As IV = 1, eq. (27) is in accordance with Einstein sponta-

neous emission formulas. We find the spectral linewidth has

been largely reduced because of the overlap integral factor

IV. In fact, a spectral line of the electronic transition from

state to ′ has been broadened to a band consisting of

numerous single spectral lines. The spectral shape function is

determined by the -function

( ).

k n kn

EE

1(

k

For an

electronic transition of given frequency

00

( )

( ))

Page 5

Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

537

there are a band of spectral lines characterized by the transi-

tion

but with different ((

.

knk n

EE

k

) k n ) and (kn) satisfying

1.4 Photon-protein resonance Raman scattering

Consider the inelastic scattering:

k photons in (

in (kn) →

(1)

k

photons in

()

k

+protein in (k′n′′).

Set

ikn

)

k

k

photons in ()

k

+ protein

() (

k

1)

k

photons in

, ,

kk

and

, 1,fk n

k

1 .

k

The scattering matrix element:

(EM)

1

E

(EM)

1

(EM)

2

.

0

fi

m

im

f Hm m H

E

i

T f Hi

i

(28)

The first term of (28) is

2

(EM)

2

0

2

0

(

1)

2

( , )

( , )d d

x

(

1)

2

( )

( )d .

kn

k n

kn

kn

e

mV

f Hi

M x Mx

e

mV

k

k

kk

kk

k

kk

kk

k

(29)

Similarly, by using

, 1,

or

, , 1 ,

III

III

mk n

k n

kk

kk

the second term of (28) is obtained:

(EM)

1

E

(EM)

1

2

2

0

;;

;;

0

(

k

1)

2

() (

)

0

( ) (

k

)

,

0

I I

k n

I

I I

k n

I

I I

k n

I

I I

k n

I

I I

k n

II I

k n

I

I I

k n

I

m

im

k n

P

kn

i

kn

k n

P

kn

kn

f Hm m H

E

i

i

e

m V

EE

EEi

k

kk

k

k

k

k

k

P

P

(30)

where

; k n

P

kn

can be simplified by eq. (8) under the

depends on I more

strongly than kInI and there are a set of resonant intermedi-

ate states with the same I but different kInI. The energy of

the resonant band {kInIIC} for a given I = IC is denoted

Condon approximation.

I I

k n

I

E

by EIC. Leaving only the resonant term in the summation,

near resonance has the following:

;;

() (

)

0

() (

)

( )

( )d

.

0

I I

k n

I

I I

k n

I

I I

k n

I

I I

k n

I

II

kn

k n

P

kn

i

kn

kn

kn IC

EE

EEi

P

P

kk

k

kk

k

P

(31)

Similarly,

;;

( ) (

k

)

0

( ) (

k

)

( )

( )d

.

0

I I

k n

I I I

k n

I

I I

k n

I

I I

k n

I

II

kn

k n

P

kn

kn

kn

knIC

EEi

EEi

P

k

k

k

k

P

P

(32)

Since

we have the completeness of the wave functions:

( )

is the solution of the eigenvalue eq. (3),

kn

( )

( )= (

).

I I

k n

IC I I

k n

IC

I I

k n

(33)

The completeness of eq. (33) has been used in the above

deduction of eqs. (31) and (32). Summing up the contribu-

tion from

1

H

and

2

,H

into eq. (28) we obtain the cross section of the quantum

transition from the initial state i to a definite final state f:

(EM)(EM)

and inserting eqs. (29)–(32)

2

0

2π

().

fik n

knfi

V

EET

kk

k

(34)

After taking the thermal average over the initial torsion

vibration states and summation over the final torsion states

and photon states in the direction d (multiplied by a factor

2

0

3

d

(2π)

V

k

k

) we obtain

2

2

2

d

d

(

1),

4π

R V

I I

e

mc

k

k

k

(35)

where IV is the TAOI given by eq. (10) or (17) and

2

2

IC

2

1(

m

) (

)

()

4

II

R

kn IC

I

EE

P

P

kk

k

(36)

or

2

2

IC

2

1(

m

) (

k

)

()

4

II

R

knIC

I

EE

P

P

k

k

(37)

is the Raman tensor and IC is the resonance width of the IC

band. Eq. (35) is in accordance with Kramers-Heisenberg

Page 6

538 Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

cross section formulas [13] as IV = 1. The factor

2

2

4π

e

mc

in

eq. (35) represents the electron classical radius. After being

excited to a higher quantum state by absorbing photons, the

orbital electron of a protein can relax to its ground state by

emitting a fluorescence photon. The inelastic cross section

eq. (35) can be used to explain the distribution and polariza-

tion of fluorescence photons. The occurrence of IV in eq.

(35) indicates that the inelastic cross section obeys the same

law of temperature dependence as in protein folding.

2 Results and discussion: test on the protein

quantum folding theory

2.1 Common factors of the thermo-averaged overlap

integral of the torsion vibration wave function

We have studied protein-photon interactions and deduced

the photon absorption/emission cross section and Raman

scattering section in protein folding. All these sections

(transition rates) have been compared with usual non-radia-

tive folding rates (without the interaction of a photon). The

general features of all photo-protein cross sections are the

proportionality of the cross section to the TAOI of the tor-

sion vibration wave function (eqs. (10), (11) and (17)). The

factor has also occurred in non-radiative folding rate for-

mulas [1]. It is the generalization of the overlap integral of

the single mode harmonic oscillators, which has been de-

scribed in previous work [14], to represent multi-modes and

non-equal frequencies between initial and final states. Since

both the initial kn and final k′n′′ are approximated by the

harmonic oscillator wave function, the overlap integral IV is

determined by two sets of harmonic frequencies {j} and

},

{

j

tween two potentials of the jth torsion modes (j = 1, …, N).

Although the overlapping wave functions in non-radiative

folding are kn and k′n′ with equal quantum number α

while in photo-folding are kn and k′n′′ with

both overlap integrals IV’s are the same functions of torsion

potential parameters j,

,

j

j and Ej. The analytical

form of IV (eq. (17)) shows how the transition rate depends

of the jth torsion potential, the

potential energy difference Ej and the angular shift j.

The overlap integral is classified into two categories:

kk

(without conformational change) and k

conformational change). The protein folding belongs to the

second category, whereas the protein photo-folding may

occur in both categories, with and without conformational

changes.

The common factor of the overlap integral IV provides

and by the energy gap Ej and angular shift j be-

,

on the frequency ratio

j

j

k

(with

important information on protein folding and photo-folding.

The detailed comparison of two kinds of folding processes

can provide evidence on the quantum nature of the folding

mechanism.

2.2 Temperature dependence of stimulated photon

emission and absorption, and resonance Raman scatter-

ing

The stimulated photon absorption and emission cross sec-

tions are given by eqs. (21) and (25), respectively. For high

incident photon flux the stimulated cross sections are large

enough to observe. The cross section of the inelastic pho-

ton-protein resonance Raman scattering is given by eq. (35).

All these protein photo-folding processes contain the same

TAOI factor IV in their cross sections, which also occurs in

the non-radiative folding rate. Since the temperature de-

pendencies of these folding rates and sections are deter-

mined by the factor IV, it indicates that the folding rates

should obey the same temperature dependence. As is well

known, the non-Arrhenius behavior of the protein folding

rate vs. temperature is a long-standing unsolved problem.

Biologists are interested in understanding why protein fold-

ing depends on temperature in such an unusual way [15].

However, from quantum folding theory, Luo and Lu have

deduced a general formula for the temperature dependence

of the non-radiative transition rate kf [3]:

2

dln

dln

1

2

,

1

T

1

T

dd

f

V

k

I

STRT

(38)

2

1,

22

B

B

k

E

k

E

SR

，

(39)

where

is a scale variable of the torsion energy, I0 represents the

average torsion inertial moment of atomic groups in the

polypeptide， N is the number of the collective torsion modes

22

0

((

jINI

22

）） (40)

of the polypeptide chain,

2

av

()

j

is the average

angular shift of the torsion potential, and

ln ln

j

j

j

N

(41)

describes the effect of the non-equal initial frequency to

final. Eq. (39) shows that the temperature dependence is

decided by three torsion potential parameters, namely the

energy gap E, the average angular shift (or in )

and the initial-to-final frequency ratio

.

In the vicinity

of the melting temperature, Tc, the temperature dependence

of E should be considered. Suppose

( )( )

E T

c

E T

Page 7

Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

539

()

c

m T

dependence in eq. (38); however, eq. (39) is replaced by

T

is near Tc. One obtains the same temperature

2

2

f

( )E T

k

( )E T

1

2

1

k T

(( )) ,

c

E T

2

cc

B

f

B

S

RG

，

(42)

where

1

( )

E T

c

c

mT

describes the structural susceptibil-

ity of the torsion potential near the melting temperature and

Gf the equilibrium folding free energy decrease measured

at temperature Tf. Note that in this case, the three torsion

potential parameters are replaced by Gf, and

Having deduced eqs. (38), (39) and (42), Luo and Lu suc-

cessfully interpreted the experimental temperature depend-

encies of all known protein folding rates.

(i) They proved that for 15 proteins whose experimental

data were available the temperature dependencies of the

folding rate are all in agreement with eq. (38).

(ii) By using S, R and the equilibrium free energy as input,

the torsion potential parameters for each protein were fully

determined in a consistent manner. The temperature de-

pendence of the folding and the unfolding rates of a protein

can be deduced in a unifying approach. It has been proved

that the mutants may have very different temperature de-

pendencies of the folding rates by virtue of the varying tor-

sion potential parameter E

that arise because of the muta-

tion [16].

Since the photo-folding and the usual non-radiative fold-

ing have the same IV factor, we can make predictions that

the temperature dependence of photo-folding obeys the

same eqs. (38)–(42), deduced from non-radiative folding.

Figures 1 and 2 give two examples. Figure 1 describes the

temperature dependence of lnkf versus 1/T for Trpcage

(PDB code 1L2Y). Figure 2 shows the dependence lnkf1/T

for the WW domain of Pin（PDB code 1PIN). In depicting

two curves the torsion potential parameters are taken from

the non-radiative folding and the experimental lnkf in

non-radiative folding at given temperatures are plotted for

reference. By taking the difference of the torsion potential

between the initial and final electronic states into account,

the torsion potential parameters E, and

( ).

c

E T

of photo-

folding are different from those in non-radiative folding,

and they are in principle dependent on the emitted or ab-

sorbed photon frequency. However, the electronic state has

an influence upon the torsion potential, mainly through E.

From eq. (39) we know that the E influences the slope

term S of lnkf 1/T relation. If the potential gap E in pho-

to-folding is greater (smaller) than that in non-radiative

folding, then the Arhenius curve of photo-folding should be

more (less) steep than Figure 1 for 1L2Y or should add a

negative (positive) correction on the slope of the curve pre-

sented in Figure 2 for 1PIN.

Note that in the deduction of the temperature dependence,

eq. (38), the “high temperature

k T

ZI

function simplification (see eq. (16)). It requires

approximation”,

2

j

2

()1

B

jj

has been assumed for the Bessel

.

j

Bj

k TI

(43)

For a typical torsion inertial moment Ij = 1037 g cm2

Figure 1 Arhenius plot for Tryptophan cage (1L2Y) folding. The param-

eters used in drawing the theoretical curve were taken as Gf = 0.7 kcal

mol1 (Tf = 296), E(Tc)=9.0kBTf, = 0.59kBTf, or = 0.12; N = 18 [3].

The experimental lnkf of the non-radiative folding at given temperatures

can be found in [17].

Figure 2 Arhenius plot for the WW domain of Pin (1PIN) folding. The

parameters used in drawing the theoretical curve are taken as Gf = 1.9

kcal mol1 (Tf = 312), E(Tc) = 43.5kBTf, = 3.85kBTf, or = 0.13; N =

99 [3]. The experimental lnkf of the non-radiative folding at given temper-

atures can be found in [17].

Page 8

540 Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

it means that

2

1.6 10 ,

j

which can be fulfilled in the

k

However, for pho-

case of conformational change,

ton emission and absorption accompanying small structural

relaxation, the protein conformational change may not occur

(i.e., kk

) and the angular shift may be small. If eq. (43)

is not fulfilled, then the original expressions (10) and (11)

for the TAOI should be used instead of eq. (17), and the

temperature dependence of the folding rate, eq. (38), should

be modified.

.k

2.3 Broadening of the spectral line and structure of the

electronic spectrum in protein photo-folding

The motion of orbital electrons obeys the wave equation, eq.

(2). In a given macromolecular configuration = 0, the

energy is

0

()

and the emitted photon frequency is

1(

()( ))

to ′. However, because of the coupling between the struc-

ture of a protein and electron motion, the electronic jump

inevitably causes protein structural relaxation or conforma-

tional changes. That is, the quantum state of the confor-

mation-electron system changes from Mkn(, x) to Mk′n′′(,

x) because of the electronic transitions. The protein struc-

ture variation, in turn, makes the frequency of the emitted

photon shift from k to

k

00

k

as the electron jumps from

k with

00

1

1

()(()( )).

knk n

EE

k

(44)

Thus the electronic transition spectrum is broadened and a

spectral band is formed that corresponds to the electronic

transition →′. The band includes the abundant vibration

spectrum without and with the conformational transition.

The former corresponds to k

sponds to .kk

The width of the spectral band is deter-

mined by the torsion vibration frequency. For example, for

the spectral line

~ 2π 10

k

order of 1013 s1, one hundredth or thousandth of the line

frequency, and it consists of a large amount of transitions

between and ′ in several tens of vibration energy levels.

Now we discuss the width of each spectral line in the

band. The rate of spontaneous emission of a photon in pro-

tein folding is given by eq. (27). It contains the overlap in-

tegral factor IV. The reason can be explained by the follow-

ing argument. If an electron jumps from one orbital to an-

other in the same molecular harmonic potential the transi-

tion will obey a strong selection rule and all transitions with

changing vibration quantum number will be forbidden be-

cause of the orthogonality of the wave functions

and

kn

().nn

However, for an electronic transi-

tion with an initial and final torsion vibration in different

k

and the latter corre-

15

s1, the band width is in the

( )

kn

'( )

harmonic potentials the vibration wave functions

and

( )

k n

cannot be orthogonal to each other and the

overlap integral exists. This is the so-called ‘forbidden’

transition. The overlap integral TAOI is an important de-

terminant factor of the ‘forbidden’ transition rate. In the

preceding calculation of a single-photon emission we esti-

mate

~10

e

s1. IV changes over a wide range. From

eqs. (17) and (18) the following is found:

( )

kn

8

VI

2

2224

0

dln

d(

1

2

1()1

.

)()2()

V

B

I

G

NIk T

(45)

This equation gives rise to IV taking a maximum at

2

()

.

NI k T

105–104. In the left of the maximum,

2

max

()

=

2

0

B

G

The typical value of

max

()

VI

is

22

max

()(),

IV

rapidly decreases as →0 following

2

const

(

exp

.

)

In

the right of the maximum, IV approximately changes with

1

() .

lifetime for an atomic energy state is about 108 s, corre-

sponding to a natural linewidth of about 6.6×108 eV.

Therefore, the width e of the spectral line in protein photo-

folding is five orders smaller than the natural linewidth.

Moreover, due to its exponential dependence on

may take a value much lower than 105 for small . This

leads to the extra-narrowness of the width e. This is a

well-marked characteristic of the photo-folding spectral lines.

Assuming IV~105 we obtain e~103 s1. A typical

2

( ) ,

IV

2.4 Conclusion

Protein photo-folding––protein folding with photon absorp-

tion or emission and the inelastic scattering of a photon on a

folded protein––is a useful field for experimentally deter-

mining whether protein folding obeys quantum laws. The

particular form of the same temperature dependence (eq.

(38)) for protein non-radiative folding and photo-folding,

and the dominant structure of the photo-folding spectral

band that consists of many narrow lines are two primary

results deduced from protein quantum folding theory. These

results are closely related to the fundamental aspects of

quantum laws. First, the results imply the existence of a set

of quantum oscillators in the transition process and these

oscillators consist primarily of the torsion-vibration type of

low frequency. Second, the results indicate that quantum

tunneling does exist in protein folding, which means the

non-locality of the state and the quantum coherence of con-

formational-electronic motion. The coherence is rooted

deeply in the cooperative motion of many structural con-

stituents (e.g., atomic electrons, molecular torsions) under a

given temperature. Experimental tests on the above predic-

tions are required and these studies will provide clearer ev-

Page 9

Luo L F. Sci China Life Sci June (2012) Vol.55 No.6

541

idence on the quantum nature of protein folding and pho-

to-folding.

The author is indebted to Drs. Lu Jun and Zhang Ying for numerous dis-

cussions on the experimental data analysis of the protein folding rates.

This work was supported by the National Natural Science Foundation of

China (Grant Nos. 202015 and 205015).

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