Highlights in Physics 200511–14 October 2005, Dipartimento di Fisica, Università di Milano

Quantum methods in protein science C. Camilloni*, P. Cerri*, D. Provasi*†, G. Tiana*† and R. A. Broglia*†

* Dipartimento di Fisica, Università di Milano† INFN – Sezione di Milano

Binding affinity: Kb

Is a measure of the effectiveness of an inhibitor F = -RT ln Kb

Metadynamics approach to quantum molecular dynamics

(A. Laio and M. Parrinello PNAS 2002 99:12562 ) :Coarse grained

History dependent (non markovian)

Active site

Metadynamics allows the exploration of free energysurface as a function of a selected set of collectivevariables (CV). A fictitious time-dependent potential acting on the CVs is added to the Lagrangian in order to escape free energy minima. In an ideal (infinite time) metdynamics simulation, after filling free energy wells, collective variables evolve in time with brownian motion. Keeping track of the hills allows the reconstruction of the free energy surface

O – N

distance (bohr)

F (Hartree)

Our unbound state has negligibletranslational entropy if compared to the translational entropy of the ligand in bulk solvent. This is a consequence of CV confinement, which we requiredfor computational reasons. We assumed that F = F(unbound)-F(bound) = F0 - TSfree + TSunbound

From our calculations

32/52/3 1

ln

N

VeRT

Finkelstein Prot. Eng. vol. 3 no. 1:1-3

Results for HCAII with trifluoromethane-sulfonamide : F0 ~ 21.1 kcal/molTS ~ 15.5 kcal/mol at room temperature (300 K)

F ~ 5.5 kcal/mol Experimental values of F are on the order of 5 kcal/mol

(A. V. Ishchenko, and E. Shakhnovich, J. Med. Chem., 45:2770)

The study of phenomena taking place in proteins that can only be described by quantum mechanics is particularly complicated, due to the large size of the system and the lack of symmetries. In these cases, a possible approach is to describe quantum-mechanically only a part of the whole protein, accounting for the rest of it in an approximated way.

• Another example of this procedure is the description of the fluorescence properties of the Green Fluorescent Protein, a protein widely used in biology to detect the expression of genes. Time dependent density functional calculations highlight the mechanism which allows to switch on and off the fluorescence by means of beams of different wavelengths.

• The work is in collaboration with the experimental group of G. Chirico (Bicocca), R. Nifosi’ (NEST, Pisa) and A. Rubio (San Sebastian).

• One example is the calculation of the binding free energy of small ligands to the Carbonic Anhydrase (HCAII), a protein involved in the glaucoma disorder. Due to the presence of a zinc atom, standard methods based on classical empirical potentials fail to predict correctly the binding free energy. The use of density functional theory, together with a novel conformational sampling algorithm, allow a description of the interaction consistent with the experimental data.

•The work is in collaboration with the groups of M. Parrinello (ETH, Lugano) and E. Shakhnovich (Harvard)

The protein can be in a fluorescent (BRIGHT) and non fluorescent (DARK) conformations, which can be switched by appropriate wavelength. The dark state corresponds to an absorption peak at 3.5 eV. To compute absorption we use the Time Dependent Density Functional Theory.

1) δ(t) perturbation to the potential;2) Calculate the 3) evolution and the density n(t);4) From the density we find the time-dependent dipole moment;5) With a Fourier transform we find the dynamic polarizability in

frequency domain and the the absorption cross section.23

0

( ) I m ( ) ( )i t ee d r x n t d t

k

21, , ,

2 effV r t r t i r tt

BRIGHT - cis

DARK - trans

Chromophore alone:We have observed the absorption peak at 3.5 eV and identified the dark state of the protein with the trans conformation of the chromophore

We have improved the degree of approximation, considering: 1) the amino acids close to the chromophore, in particular the hydrogens bond network, 2) the chromophore and the Coulombian field of the whole protein, showing that the two effects are complementary on the absorption spectrum prediction and must be accounted in a realistic description of the optical properties of the system.

Collective variable (s)

Fre

e en

ergy

F(s

)

unbound

bound

These results are obtained including effects due to the electrostatic field generated by the part of theprotein not included in our ab-initio calculations. We observed that neglecting these effects yields wrong results

Zn – N distance (bohr)