Protein Photo-folding and Quantum Folding Theory

The rates of protein folding with photon absorption or emission and the cross section of photon -protein inelastic scattering are calculated from the quantum folding theory by use of standard field-theoretical method. All these protein photo-folding processes are compared with common protein folding without interaction of photons (nonradiative folding). It is demonstrated that there exists a common factor (thermo-averaged overlap integral of vibration wave function, TAOI) for protein folding and protein photo-folding. Based on this finding it is predicted that: 1) the stimulated photo-folding rates show the same temperature dependence as protein folding; 2) the spectral line of electronic transition is broadened to a band which includes abundant vibration spectrum without and with conformational transition and the width of the vibration spectral line is largely reduced; 3) the resonance fluorescence cross section changes with temperature obeying the same law (Luo-Lu's law). The particular form of the folding rate - temperature relation and the abundant spectral structure imply the existence of a set of quantum oscillators in the transition process and these oscillators are mainly of torsion type of low frequency, imply the quantum tunneling between protein conformations does exist in folding and photo-folding processes and the tunneling is rooted deeply in the coherent motion of the conformational-electronic system.


I Introduction
Protein is a microscopic system of atoms and molecules. From modern physics it should obey quantum laws in principle. Recently we put forward a protein quantum folding theory [1] [2]. Although the bioinformatics studies such as the prediction of protein structure and function from molecular sequence has achieved great successes the dynamical problem on protein folding remains unclear and to be solved. The proposed quantum folding theory emphasizes the idea of torsion cooperative transition. The importance of torsion state can be looked as follows: as a multi-atom system, the conformation of a protein is fully determined by bond lengths, bond angles, and torsion angles (dihedral angles), among which the torsion angles are most easily changed even at room temperature and usually assumed as the main variables of protein conformation. Simultaneously, the torsion potential generally has several minima the transition between which is responsible for the conformational change. All torsion modes between contact residues are taken into account in quantum folding theory. These modes are assumed to participate in the quantum transition cooperatively. About the cooperativeness the Bose condensation of strongly exited longitudinal electric modes of living system was proposed as early as in the seventies of last century [3]. The fold cooperativeness of a protein was also demonstrated in earlier literatures [4]. These authors explained the possible existence of the cooperativeness in protein folding or in living system from the point of non-linear dynamics and thermodynamics. In the meantime, the contact order was introduced as an important parameter for understanding and quantitatively describing folding rate [5]. Recently, the dihedral transition was observed more directly in statistical analysis of protein conformational changes [6]. They indicated the cooperative dihedral transitions occur in most (about 82%) polypeptide chain. Based on above considerations the nonradiative quantum folding theory is formulated. The theory has successfully explained the non-Arrhenius behavior of the temperature dependence of protein folding rates [1] [2]. Figure 1 gives an intuitive example, it gives the schematic diagram of the cooperative transition of all torsion angles of protein 1enh (Engrailed Homeo domain).
To explore the fundamental physics behind the folding more deeply and clarify the quantum nature of the folding mechanism more clearly we shall study the protein photo-folding processes, namely, the photon emission or absorption in protein folding and the inelastic scattering of photon on protein (photon-protein resonance Raman scattering). Although the fluorescence technique has been largely developed in recent years and widely employed in studying protein folding and protein-protein interaction dynamics [7][8] the theoretical calculation of fluorescence and other protein photo-folding processes from fundamental interaction seems have not been found in literatures. The situation may be attributed to our "too-classical" understanding of protein folding. But the new-found quantum folding theory affords a sound basis for studying these problems. In the theory the photon emission or absorption in protein folding and the inelastic scattering of photon on protein, as electromagnetic processes, can be accurately described by quantum electrodynamics. The emission or absorption of a photon by the atomic electron is the first step. Then, due to the electronic transition emitting or absorbing photon is coupled to the conformational change of protein structure, the torsion transition in polypeptide chain plays an important role in determining the photon emission /absorption rates or cross sections. We shall make the first-principle-calculation of the rates and cross sections of these processes based on quantum transition theory. The quantitative results provide further checkpoints on the quantum folding theory. It includes: the temperature dependence of the stimulated protein photo-folding rates, the structure and width of the spectral lines of electronic transition in protein-folding, and the temperature dependence of the resonance fluorescence cross section. The experimental tests of these theoretical predictions will support the idea on quantum coherence of the electronicconformational transition and afford more clear evidences on the quantum nature of protein folding and photo-folding. , i j χ is defined through side chain 4 atoms in the same manner. The total number of dihedral angles in this example is 23 which are assumed to participate in the quantum transition cooperatively. In protein photo-folding the atomic electron jumps from α to α' emitting or absorbing a photon and the electronic transition is coupled to the torsion transition. Studying protein photo-folding gives an efficient way to demonstrate the quantum nature of protein folding.
H contains electronic kinetic energy term, from gauge invariance of we obtain electromagnetic interaction where m is electron mass, c λ

Stimulated photon emission and absorption in protein folding
We discuss single photon absorption at first. Set α α P is the matrix element of electron momentum. In the above deduction of Eq (8) the Condon approximation, namely, the matrix element not dependent on θ has been used. Assuming the torsion potential (a term occurring in  (14) ( , ) B n T is the Boltzmann factor for thermal average, j I is inertial moment of the j-th torsion mode , j δθ is the angular displacement, and j E δ is the energy gap between the initial and final states for the j-th mode. p j represents the net change in quantum number for oscillator mode j, which satisfies the constraint in the summation of Eq. (10) . V I is called Thermo-Averaged Overlap Integral (TAOI). By use of the asymptotic formula for Bessel function [9] for z>>1 ( ω is the average of initial torsion frequencies j ω over oscillator mode j ). E Δ is a potential parameter, the gap of the energy minimum between initial and final torsion state. In Eq (17) the energy gap E Δ has been replaced by G Δ to take the frequency difference ' j j ω ω ≠ into account. The detailed deduction of V I can be found in [1]. It is a function of j ω (or its average ω ), ' j ω (or its average ω '), 2 j δθ (or its average ( δθ ) 2 )and j E δ (or its sum E Δ ). Notice that the simplified expression (17) is obtained when z j >>1 . For photo-folding with conformational change, ' k k ≠ , the condition is fulfilled generally. The single photon absorption cross section is obtained readily from (9) By comparison with nonradiative folding rate [1] is the average inertial moment and j l means the magnetic quantum number of electronic wave function ( , ) x α ϕ θ ) we obtain the ratio of rates where F is the photon flux. Setting The double and multi-photon stimulated emission can be calculated through the second and the higher order perturbation. All results contain TAOI factor V I .

Spontaneous emission in protein folding and the spectral line structure of photo-folding
Following the same perturbation approach and setting initial λ ν k =0 in the above deduction of stimulated emission one obtains the single-photon spontaneous emission rate. The rate of quantum transition from a given initial state i knα = to the definite final state Considering that in spontaneous emission case no stimulating electromagnetic field exists and the frequency of emitted photon is not given a priori. Adopting continuous representation of electromagnetic field expansion and replacing the sum over photon final states

Photon-protein resonance Raman scattering
For inelastic scattering, set The scattering matrix element The first term of (28) is Similarly, by using  I I I  I I I  I I I  I I I   I I I  I I I  I I

Results and Discussions: Test on Protein Quantum Folding Theory Common factor of thermo-averaged overlap integral of torsion vibration wave function
We have studied the protein-photon interaction 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 nonradiative folding rates (without interaction of photon). The common features of all photo-protein cross sections are the proportionality of cross section to TAOI V I of torsion vibration wave function (Eqs (10), (11) and (17)). The factor has also occurred in nonradiative folding rate formulas [1]. It is the generalization of the overlap integral of single mode harmonic oscillators in previous work [11] to the case of multi-modes and non-equal frequencies between initial and final states. Since both initial

First test -Temperature dependence of stimulated photon emission and absorption
The stimulated photon absorption and emission cross sections are given by Eq (20) and (25) respectively. For high incident photon flux the stimulated cross sections are large enough for observation. Since the temperature dependencies of these folding rates and sections are fully determined by factor V I it leads to a conclusion that they should obey the same temperature relation as nonradiative protein folding. Luo and Lu have deduced a general formula for the temperature dependence of nonradiative transition rate W [2], 2 ln ln . Now we predict the same temperature dependence (Luo-Lu's law) also holds for photo-folding processes. Since the result is deduced from quantum folding theory it provides a checkpoint for this theory. The experimental test will furnish new evidence on the theory and the view of protein folding as a quantum transition.  (10) and (11) for TAOI should be used instead of Eq (17). In this case, the temperature dependence is more complicate than given by Eqs (38)

Third test -Temperature dependence of resonance fluorescence cross section
The cross section of inelastic photon-protein resonance Raman scattering is given by Eqs (35)-(37). Due to the TAOI factor I V the inelastic cross section behaves the same temperature dependence as in protein foding rates. Non-Arrhenius behavior, Eq (38), should be observed in experiments.
The general relation Eq (38) of temperature dependence is a characteristic of torsion quantum oscillators. To clarify this point we consider the vibration of bond length and bond angle (stretching and bending) for comparison. Of course, both the bond length and the bond angle of macromolecule are important dynamical variables and their vibration can be put into the formalism of quantum theory. However, their temperature dependence should exhibit another characteristics different from Eq (38). The reason is twofold. First, due to the frequency difference between two kinds of modes − torsion and stretch /bend, the Boson condensation occurs only in torsion but not generally in stretch or bend modes. Consider the vibration partition function of a molecule in harmonic conformational potential.
By means of (45) one finds the strong dependence of S on frequency ω. For example, at T = 14 300 K as the frequency 2 ω ν π = is lowered down to 10 13 Hz, the entropy increases rapidly with the decrease of ν , at , respectively. So, the Boson condensation occurs and the free energy of the oscillator system drops for torsion vibrations of low frequency. There is a large gap of free energy between two kinds of oscillators. Second, for bond stretching or bending, due to lack of multi-minima in the potential only the protein structural relaxation can occur in electronic transition and no large conformational change can be defined. So the temperature dependence of stretching and bending oscillation should exhibit characteristics different from Eq (38).  25)). This is an advantage of protein photo-folding in theoretical studies. Thus, the comparative study of radiation-less folding and photo-folding can give more information on protein folding, including the information on the electronic wave function factor E I (or A ) of the nonradiative folding.
Although the condition of α =α ' holds for most nonradiative folding, how the result changes if the condition is not assumed? The general form of nonradiative transition matrix element is given by [1] converted to its excited state and can then propagate the excitation further. TAOI as a measure of the correlation of two quantum oscillators can be introduced in this problem and the subject about the long-distance energy and information transfer between microscopic constituents of living system can be studied in this approach.