Implementation of a symplectic multiple-time-step molecular dynamics algorithm, based on the united-residue mesoscopic potential energy function

A symplectic multiple-time-step (MTS) algorithm has been developed for the united-residue (UNRES) force field. In this algorithm, the slow-varying forces (which contain most of the long-range interactions and are, therefore, expensive to compute) are integrated with a larger time step, termed the basic time step, and the fast-varying forces are integrated with a shorter time step, which is an integral fraction of the basic time step. Based on the split operator formalism, the equations of motion were derived. Separation of the fast- and slow-varying forces leads to stable molecular dynamics with longer time steps. The algorithms were tested with the Ala(10) polypeptide chain and two versions of the UNRES force field: the current one in which the energy components accounting for the energetics of side-chain rotamers (U(rot)) can lead to numerically unstable forces and a modified one in which the the present U(rot) was replaced by a numerically stable expression which, at present, is parametrized only for polyalanine chains. With the modified UNRES potential, stable trajectories were obtained even when extending the basic time step to 15 fs and, with the original UNRES potentials, the basic time step is 1 fs. An adaptive multiple-time-step (A-MTS) algorithm is proposed to handle instabilities in the forces; in this method, the number of substeps in the basic time step varies depending on the change of the magnitude of the acceleration. With this algorithm, the basic time step is 1 fs but the number of substeps and, consequently, the computational cost are reduced with respect to the MTS algorithm. The use of the UNRES mesoscopic energy function and the algorithms derived in this work enables one to increase the simulation time period by several orders of magnitude compared to conventional atomic-resolution molecular dynamics approaches and, consequently, such an approach appears applicable to simulating protein-folding pathways, protein functional dynamics in a real molecular environment, and dynamical molecular recognition processes.     

Molecular dynamics with the united-residue model of polypeptide chains. II. Langevin and Berendsen-bath dynamics and tests on model alpha-helical systems

The implementation of molecular dynamics (MD) with our physics-based protein united-residue (UNRES) force field, described in the accompanying paper, was extended to Langevin dynamics. The equations of motion are integrated by using a simplified stochastic velocity Verlet algorithm. To compare the results to those with all-atom simulations with implicit solvent in which no explicit stochastic and friction forces are present, we alternatively introduced the Berendsen thermostat. Test simulations on the Ala(10) polypeptide demonstrated that the average kinetic energy is stable with about a 5 fs time step. To determine the correspondence between the UNRES time step and the time step of all-atom molecular dynamics, all-atom simulations with the AMBER 99 force field and explicit solvent and also with implicit solvent taken into account within the framework of the generalized Born/surface area (GBSA) model were carried out on the unblocked Ala(10) polypeptide. We found that the UNRES time scale is 4 times longer than that of all-atom MD simulations because the degrees of freedom corresponding to the fastest motions in UNRES are averaged out. When the reduction of the computational cost for evaluation of the UNRES energy function is also taken into account, UNRES (with hydration included implicitly in the side chain-side chain interaction potential) offers about at least a 4000-fold speed up of computations relative to all-atom simulations with explicit solvent and at least a 65-fold speed up relative to all-atom simulations with implicit solvent. To carry out an initial full-blown test of the UNRES/MD approach, we ran Berendsen-bath and Langevin dynamics simulations of the 46-residue B-domain of staphylococcal protein A. We were able to determine the folding temperature at which all trajectories converged to nativelike structures with both approaches. For comparison, we carried out ab initio folding simulations of this protein at the AMBER 99/GBSA level. The average CPU time for folding protein A by UNRES molecular dynamics was 30 min with a single Alpha processor, compared to about 152 h for all-atom simulations with implicit solvent. It can be concluded that the UNRES/MD approach will enable us to carry out microsecond and, possibly, millisecond simulations of protein folding and, consequently, of the folding process of proteins in real time.     

Implementations of Nosé-Hoover and Nosé-Poincaré thermostats in mesoscopic dynamic simulations with the united-residue model of a polypeptide chain

Molecular dynamics (MD) simulations generate a canonical ensemble only when integration of the equations of motion is coupled to a thermostat. Three extended phase space thermostats, one version of Nose-Hoover and two versions of Nose-Poincare, are compared with each other and with the Berendsen thermostat and Langevin stochastic dynamics. Implementation of extended phase space thermostats was first tested on a model Lennard-Jones fluid system; subsequently, they were implemented with our physics-based protein united-residue (UNRES) force field MD. The thermostats were also implemented and tested for the multiple-time-step reversible reference system propagator (RESPA). The velocity and temperature distributions were analyzed to confirm that the proper canonical distribution is generated by each simulation. The value of the artificial mass constant, Q, of the thermostat has a large influence on the distribution of the temperatures sampled during UNRES simulations (the velocity distributions were affected only slightly). The numerical stabilities of all three algorithms were compared with each other and with that of microcanonical MD. Both Nose-Poincare thermostats, which are symplectic, were not very stable for both the Lennard-Jones fluid and UNRES MD simulations started from nonequilibrated structures which implies major changes of the potential energy throughout a trajectory. Even though the Nose-Hoover thermostat does not have a canonical symplectic structure, it is the most stable algorithm for UNRES MD simulations. For UNRES with RESPA, the "extended system inside-reference system propagator algorithm" of the RESPA implementation of the Nose-Hoover thermostat was the only stable algorithm, and enabled us to increase the integration time step.     

Ewald summation and multiple time step methods for molecular dynamics simulation of biological molecules

Methods by which to determine conditions for a molecular dynamics (MD) simulation of biological molecules were investigated. Derivation of the optimal parameters of the Ewald summation was described so as to give same precision to the real space, the reciprocal space summations and the van der Waals interaction. Later, the procedure by which to determine the condition of the multiple time step method by RESPA (REference System Propagator Algorithm; Tuckerman et al., 1992, J. Chem. Phys., 97, 1990) was described as exemplified by MD simulations of a solvated β-sheet peptide. The conservation of the total energy in a microcanonical ensemble was measured to investigate the stability of the simulation conditions. The most feasible respective combinations of the time steps were: 0.25 fs for bond, angle and torsion interactions; 2 fs for van der Waals interaction and Ewald real-space summation; and 4 fs for Ewald reciprocal-space summation. Though it retained an acceptable accuracy, this condition accelerated the simulation ten-fold compared to that in which a simple velocity-Verlet method with a time step of 0.25 fs was used. The update of the correction term due to excluded neighbors was then investigated. Better results were obtained when the correction was updated with the real-space than when it was updated with the reciprocal-space summation. Finally, an MD simulation as long as 50 ps performed under the optimal Ewald and RESPA parameters was thus determined. The trajectory showed a good stability, indicating the feasibility of the parameters.     

Evaluation of atomic pressure in the multiple time‐step integration algorithm

In molecular dynamics (MD) calculations, reduction in calculation time per MD loop is essential. A multiple time‐step (MTS) integration algorithm, the RESPA (Tuckerman and Berne, J. Chem. Phys. 1992, 97, 1990–2001), enables reductions in calculation time by decreasing the frequency of time‐consuming long‐range interaction calculations. However, the RESPA MTS algorithm involves uncertainties in evaluating the atomic interaction‐based pressure (i.e., atomic pressure) of systems with and without holonomic constraints. It is not clear which intermediate forces and constraint forces in the MTS integration procedure should be used to calculate the atomic pressure. In this article, we propose a series of equations to evaluate the atomic pressure in the RESPA MTS integration procedure on the basis of its equivalence to the Velocity‐Verlet integration procedure with a single time step (STS). The equations guarantee time‐reversibility even for the system with holonomic constrants. Furthermore, we generalize the equations to both (i) arbitrary number of inner time steps and (ii) arbitrary number of force components (RESPA levels). The atomic pressure calculated by our equations with the MTS integration shows excellent agreement with the reference value with the STS, whereas pressures calculated using the conventional ad hoc equations deviated from it. Our equations can be extended straightforwardly to the MTS integration algorithm for the isothermal NVT and isothermal–isobaric NPT ensembles. © 2017 Wiley Periodicals, Inc.     

Difficulties with multiple time stepping and fast multipole algorithm in molecular dynamics

Numerical experiments are performed on a 36,000-atom protein–DNA–water simulation to ascertain the effectiveness of two devices for reducing the time spent computing long-range electrostatics interactions. It is shown for Verlet-I/r-RESPA multiple time stepping, which is based on approximating long-range forces as widely separated impulses, that a long time step of 5 fs results in a dramatic energy drift and that this is reduced by using an even larger long time step. It is also shown that the use of as many as six terms in a fast multipole algorithm approximation to long-range electrostatics still fails to prevent significant energy drift even though four digits of accuracy is obtained. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1785–1791, 1997     

Modification and optimization of the united-residue (UNRES) potential energy function for canonical simulations. I. Temperature dependence of the effective energy function and tests of the optimization method with single training proteins

We report the modification and parametrization of the united-residue (UNRES) force field for energy-based protein structure prediction and protein folding simulations. We tested the approach on three training proteins separately: 1E0L (beta), 1GAB (alpha), and 1E0G (alpha + beta). Heretofore, the UNRES force field had been designed and parametrized to locate native-like structures of proteins as global minima of their effective potential energy surfaces, which largely neglected the conformational entropy because decoys composed of only lowest-energy conformations were used to optimize the force field. Recently, we developed a mesoscopic dynamics procedure for UNRES and applied it with success to simulate protein folding pathways. However, the force field turned out to be largely biased toward -helical structures in canonical simulations because the conformational entropy had been neglected in the parametrization. We applied the hierarchical optimization method, developed in our earlier work, to optimize the force field; in this method, the conformational space of a training protein is divided into levels, each corresponding to a certain degree of native-likeness. The levels are ordered according to increasing native-likeness; level 0 corresponds to structures with no native-like elements, and the highest level corresponds to the fully native-like structures. The aim of optimization is to achieve the order of the free energies of levels, decreasing as their native-likeness increases. The procedure is iterative, and decoys of the training protein(s) generated with the energy function parameters of the preceding iteration are used to optimize the force field in a current iteration. We applied the multiplexing replica-exchange molecular dynamics (MREMD) method, recently implemented in UNRES, to generate decoys; with this modification, conformational entropy is taken into account. Moreover, we optimized the free-energy gaps between levels at temperatures corresponding to a predominance of folded or unfolded structures, as well as to structures at the putative folding-transition temperature, changing the sign of the gaps at the transition temperature. This enabled us to obtain force fields characterized by a single peak in the heat capacity at the transition temperature. Furthermore, we introduced temperature dependence to the UNRES force field; this is consistent with the fact that it is a free-energy and not a potential energy function. beta     

MOIL-opt: Energy-Conserving Molecular Dynamics on a GPU/CPU system

We report an optimized version of the molecular dynamics program MOIL that runs on a shared memory system with OpenMP and exploits the power of a Graphics Processing Unit (GPU). The model is of heterogeneous computing system on a single node with several cores sharing the same memory and a GPU. This is a typical laboratory tool, which provides excellent performance at minimal cost. Besides performance, emphasis is made on accuracy and stability of the algorithm probed by energy conservation for explicit-solvent atomically-detailed-models. Especially for long simulations energy conservation is critical due to the phenomenon known as "energy drift" in which energy errors accumulate linearly as a function of simulation time. To achieve long time dynamics with acceptable accuracy the drift must be particularly small. We identify several means of controlling long-time numerical accuracy while maintaining excellent speedup. To maintain a high level of energy conservation SHAKE and the Ewald reciprocal summation are run in double precision. Double precision summation of real-space non-bonded interactions improves energy conservation. In our best option, the energy drift using 1fs for a time step while constraining the distances of all bonds, is undetectable in 10ns simulation of solvated DHFR (Dihydrofolate reductase). Faster options, shaking only bonds with hydrogen atoms, are also very well behaved and have drifts of less than 1kcal/mol per nanosecond of the same system. CPU/GPU implementations require changes in programming models. We consider the use of a list of neighbors and quadratic versus linear interpolation in lookup tables of different sizes. Quadratic interpolation with a smaller number of grid points is faster than linear lookup tables (with finer representation) without loss of accuracy. Atomic neighbor lists were found most efficient. Typical speedups are about a factor of 10 compared to a single-core single-precision code.     

Coarse-grained force field: general folding theory

We review the coarse-grained UNited RESidue (UNRES) force field for the simulations of protein structure and dynamics, which is being developed in our laboratory over the last several years. UNRES is a physics-based force field, the prototype of which is defined as a potential of mean force of polypeptide chains in water, where all the degrees of freedom except the coordinates of α-carbon atoms and side-chain centers have been integrated out. We describe the initial implementation of UNRES to protein-structure prediction formulated as a search for the global minimum of the potential-energy function and its subsequent molecular dynamics and extensions of molecular-dynamics implementation, which enabled us to study protein-folding pathways and thermodynamics, as well as to reformulate the protein-structure prediction problem as a search for the conformational ensemble with the lowest free energy at temperatures below the folding-transition temperature. Applications of UNRES to study biological problems are also described.     

Olefin epoxidation with tert-butyl hydroperoxide catalyzed by MoO2X2L complexes: a DFT mechanistic study

DFT calculations suggest that the catalytic epoxidation of olefins by Mo(vi) complexes, modeled by MoO2Br2(MeN=C(H)-C(H)=NMe), in the presence of MeOOH, the model for tert-butyl hydroperoxide, starts with a hydrogen transfer from the peroxide to one of the terminal Mo=O oxygen atoms and the remaining MeOO anion binds as a seventh ligand, forming a five-membered Mo-O(alpha)-O(beta)(Me)...H-O-Mo ring held together by a hydrogen bond. In the second step, a concerted approach of ethylene to the Mo-O(alpha) bond gives rise to an intermediate containing a seven-membered Mo-C-C-O(alpha)-O(beta)(Me)...H-O-Mo ring. In the final step, decomposition of the intermediate leads to the starting complex, alcohol and the epoxide. The activation energy for the addition of the olefin (second step) is the highest one, in agreement with available kinetic studies showing that the catalyst formation is not always a rate-limiting step. DFT calculations also show that the alcohol by-product (MeOH) can react with the starting complex, competing with ROOH and hence leading to the progressive catalyst poisoning, which has been observed experimentally.     

Probing the role of the C-H...O hydrogen bond stabilized polypeptide chain reversal at the C-terminus of designed peptide helices. Structural characterization of three decapeptides

The structural characterization in crystals of three designed decapeptides containing a double d-segment at the C-terminus is described. The crystal structures of the peptides Boc-Leu-Aib-Val-Xxx-Leu-Aib-Val-(D)Ala-(D)Leu-Aib-OMe, (Xxx = Gly 2, (D)Ala 3, Aib 4) have been determined and compared with those reported earlier for peptide 1 (Xxx = Ala) and the all l analogue Boc-Leu-Aib-Val-Ala-Leu-Aib-Val-Ala-Leu-Aib-OMe, which yielded a perfect right-handed alpha-helical structure. Peptides 1 and 2 reveal a right-handed helical segment spanning residues 1 to 7, ending in a Schellman motif with (D)Ala(8) functioning as the terminating residue. Polypeptide chain reversal occurs at residue 9, a novel feature that appears to be the consequence of a C-H.O hydrogen bond between residue 4 C(alpha)H and residue 9 CO groups. The structures of peptides 3 and 4, which lack the pro R hydrogen at the C(alpha) atom of residue 4, are dramatically different. Peptide 3 adopts a right-handed helical conformation over the 1 to 7 segment. Residues 8 and 9 adopt alpha(L) conformations forming a C-terminus type I' beta-turn, corresponding to an incipient left-handed twist of the polypeptide chain. In peptide 4, helix termination occurs at Aib(6), with residues 6 to 9 forming a left-handed helix, resulting in a structure that accommodates direct fusion of two helical segments of opposite twist. Peptides 3 and 4 provide examples of chiral residues occurring in the less favored sense of helical twist; (D)Ala(4) in peptide 3 adopts an alpha(R) conformation, while (L)Val(7) in 4 adopts an alpha(L) conformation. The structural comparison of the decapeptides reported here provides evidence for the role of specific C-H.O hydrogen bonds in stabilizing chain reversals at helix termini, which may be relevant in aligning contiguous helical and strand segments in polypeptide structures.     

Increasing the time step with mass scaling in Born‐Oppenheimer ab initio QM/MM molecular dynamics simulations

Born‐Oppenheimer ab initio QM/MM molecular dynamics simulation with umbrella sampling is a state‐of‐the‐art approach to calculate free energy profiles of chemical reactions in complex systems. To further improve its computational efficiency, a mass‐scaling method with the increased time step in MD simulations has been explored and tested. It is found that by increasing the hydrogen mass to 10 amu, a time step of 3 fs can be employed in ab initio QM/MM MD simulations. In all our three test cases, including two solution reactions and one enzyme reaction, the resulted reaction free energy profiles with 3 fs time step and mass scaling are found to be in excellent agreement with the corresponding simulation results using 1 fs time step and the normal mass. These results indicate that for Born‐Oppenheimer ab initio QM/MM molecular dynamics simulations with umbrella sampling, the mass‐scaling method can significantly reduce its computational cost while has little effect on the calculated free energy profiles. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Explicit symplectic integrators of molecular dynamics algorithms for rigid-body molecules in the canonical, isobaric-isothermal, and related ensembles

The authors propose explicit symplectic integrators of molecular dynamics (MD) algorithms for rigid-body molecules in the canonical and isobaric-isothermal ensembles. They also present a symplectic algorithm in the constant normal pressure and lateral surface area ensemble and that combined with the Parrinello-Rahman algorithm. Employing the symplectic integrators for MD algorithms, there is a conserved quantity which is close to Hamiltonian. Therefore, they can perform a MD simulation more stably than by conventional nonsymplectic algorithms. They applied this algorithm to a TIP3P pure water system at 300 K and compared the time evolution of the Hamiltonian with those by the nonsymplectic algorithms. They found that the Hamiltonian was conserved well by the symplectic algorithm even for a time step of 4 fs. This time step is longer than typical values of 0.5-2 fs which are used by the conventional nonsymplectic algorithms.     

Multiple time scale molecular dynamics for fluids with orientational degrees of freedom. II. Canonical and isokinetic ensembles

We have developed several multiple time stepping techniques to overcome the limitations on efficiency of molecular dynamics simulations of complex fluids. They include the modified canonical and isokinetic schemes, as well as the extended isokinetic Nosé-Hoover chain approach. The latter generalizes the method of Minary, Tuckerman, and Martyna for translational motion [Phys. Rev. Lett. 93, 150201 (2004)] to systems with both translational and orientational degrees of freedom. Although the microcanonical integrators are restricted to relatively small outer time steps of order of 16 fs, we show on the basis of molecular dynamics simulations of ambient water that in the canonical and isokinetic thermostats the size of these steps can be increased to 50 and 75 fs, respectively (at the same inner time step of 4 fs). Within the generalized isokinetic Nosé-Hoover chain algorithm we have derived, huge outer time steps of order of 500 fs can be used without losing numerical stability and affecting equilibrium properties.     

Electron and nuclear dynamics of molecular clusters in ultraintense laser fields. II. Electron dynamics and outer ionization of the nanoplasma

We explore electron dynamics in molecular (CD4)(1061) clusters and elemental Xen (n=249-2171) clusters, responding to ultraintense (intensity I=10(16)-10(19) W cm(-2)) laser fields. Molecular dynamics simulations (including magnetic field and relativistic effects) and analyses of high-energy electron dynamics and nuclear ion dynamics in a cluster interacting with a Gaussian shaped laser field (frequency 0.35 fs(-1), photon energy 1.44 eV, phase 0, temporal width 25 fs) elucidated the time dependence of inner ionization, the formation of a nanoplasma of unbound electrons within the cluster or its vicinity, and of outer ionization. We determined the cluster size and the laser intensity dependence of these three sequential-parallel electronic processes. The characteristic times for cluster inner ionization (tau(ii)) and for outer ionization (tau(oi)) fall in the femtosecond time domain, i.e., tau(ii)=2-9 fs and tau(oi)=4-15 fs for (CD4)(1061), tau(ii)=7-30 fs and tau(oi)=5-13 fs for Xe(n) (n=479,1061), with both tau(ii) and tau(oi) decreasing with increasing I, in accord with the barrier suppression ionization mechanism for inner ionization of the constituents and the cluster barrier suppression ionization mechanism for outer ionization. The positive delay times Deltatau(OI) between outer and inner ionization (e.g., Deltatau(OI)=6.5 fs for Xen at I=10(16) W cm(-2) and Deltatau(OI)=0.2 fs for (CD4)(1061) at I=10(19) W cm(-2)) demonstrate that the outer/inner ionization processes are sequential. For (CD4)(1061), tau(ii)tau(oi), reflecting on the energetic hierarchy in the ionization of the Xe atoms. Quasiresonance contributions to the outer ionization of the nanoplasma were established, as manifested in the temporal oscillations in the inner/outer ionization levels, and in the center of mass of the nanoplasma electrons. The formation characteristics, dynamics, and response of the nanoplasma in molecular or elemental clusters were addressed. The nanoplasma is positively charged, with a high-average electron density [rho(P)=(2-3)10(22) cm(-3)], being characterized by high-average electron energies epsilon(av) (e.g., in Xe(1061) clusters epsilon(av)=54 eV at I=10(16) W cm(-2) and epsilon(av)=0.56-0.37 keV at I=10(18) W cm(-2), with epsilon(av) proportional, variant I(1/2)). Beyond the cluster boundary the average electron energy markedly increases, reaching electron energies in the range of 1.2-40 keV for outer ionization of Xe(n) (n=249-2171) clusters. The nanoplasma exhibits spatial inhomogeneity and angular anisotropy induced by the laser field. Femtosecond time scales are predicted for the nanoplasma production (rise times 7-3 fs), for the decay (decay times approximately 5 fs), and for the persistence time (30-10 fs) of a transient nanoplasma at I=10(17)-10(18) W cm(-2). At lower intensities of I=10(16) W cm(-2) a persistent nanoplasma with a "long" lifetime of > 50 fs will prevail.     

Computationally efficient canonical molecular dynamics simulations by using a multiple time‐step integrator algorithm combined with the particle mesh Ewald method and with the fast multipole method

An efficient implementation of the canonical molecular dynamics simulation using the reversible reference system propagator algorithm (r‐RESPA) combined with the particle mesh Ewald method (PMEM) and with the macroscopic expansion of the fast multipole method (MEFMM) was examined. The performance of the calculations was evaluated for systems with 3000, 9999, 30,000, 60,000, and 99,840 particles. For a given accuracy, the optimal conditions for minimizing the CPU time for the implementation of the Ewald method, the PMEM, and the MEFMM were first analyzed. Using the optimal conditions, we evaluated the performance and the reliability of the integrated methods. For all the systems examined, the r‐RESPA with the PMEM was about twice as fast as the r‐RESPA with the MEFMM. The difference arose from the difference in the numerical complexities of the fast Fourier transform in the PMEM and from the transformation of the multipole moments into the coefficients of the local field expansion in the MEFMM. Compared with conventional methods, such as the velocity‐verlet algorithm with the Ewald method, significant speedups were obtained by the integrated methods; the speedup of the calculation was a function of system size, and was a factor of 100 for a system with 3000 particles and increased to a factor of 700 for a system with 99,840 particles. These integrated calculations are, therefore, promising for realizing large‐scale molecular dynamics simulations for complex systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 201–217, 2000     

Polymer dynamics in torsion space

The large scale motion of proteins, or covalently bonded polymers in general, is governed by the dynamics of the torsion angles, with bond lengths and bond angles kept approximately constant. In the present work, the Lagrangian equations of torsion motion are derived for a general macromolecule. The dynamics is implemented numerically for a test protein, using the velocity Verlet method as the integrator. The results indicate time steps of up to about 30 fs can be used for short time (up to at least 20 ps) simulations, before the dynamics and energy start to differ significantly from results obtained with smaller time steps. For longer time simulations, up to 1000 ps, a time step of 10 fs is relatively safe. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Balancing an accurate representation of the molecular surface in generalized born formalisms with integrator stability in molecular dynamics simulations

Different integrator time steps in NVT and NVE simulations of protein and nucleic acid systems are tested with the GBMV (Generalized Born using Molecular Volume) and GBSW (Generalized Born with simple SWitching) methods. The simulation stability and energy conservation is investigated in relation to the agreement with the Poisson theory. It is found that very close agreement between generalized Born methods and the Poisson theory based on the commonly used sharp molecular surface definition results in energy drift and simulation artifacts in molecular dynamics simulation protocols with standard 2-fs time steps. New parameters are proposed for the GBMV method, which maintains very good agreement with the Poisson theory while providing energy conservation and stable simulations at time steps of 1 to 1.5 fs.     

Ultrafast fluorescence studies of dye sensitized solar cells

Time-resolved fluorescence spectra from the RuN719 dye exhibit very short lifetimes (<30 fs) in solutions, on non-injecting substrates and on injecting ones. This reveals <10 fs intramolecular energy redistribution competing with the injection. We conclude that injection proceeds on a sub-10 fs time scale from non-thermalized levels of the dye.