Optimizing transition interface sampling simulations

We demonstrate that a recently proposed adaptive optimization algorithm for forward flux sampling simulations [E. E. Borrero and F. A. Escobedo, J. Chem. Phys. 129, 024115 (2008)] can be easily applied within the framework of transition interface sampling. This optimization algorithm systematically identifies the kinetic bottlenecks along the order parameter used to partition phase space via interfaces and improves the statistical accuracy of the reaction rate constant estimate. In different versions of the algorithm, the number or the placement of the interfaces (or both) are varied in order to allocate the numerical effort in a balanced way. The algorithm is demonstrated for a simple two-dimensional model and for the dipole flip transition of icelike structures inside carbon nanotubes. For these test systems, the optimization yielded an efficiency increase by a factor of 2-15.     

An efficient self-optimized sampling method for rare events in nonequilibrium systems

Rare events such as nucleation processes are of ubiquitous importance in real systems.The most popular method for nonequilibrium systems,forward flux sampling(FFS),samples rare events by using interfaces to partition the whole transition process into sequence of steps along an order parameter connecting the initial and final states.FFS usually suffers from two main difficulties:low computational efficiency due to bad interface locations and even being not applicable when trapping into unknown intermediate metastable states.In the present work,we propose an approach to overcome these difficulties,by self-adaptively locating the interfaces on the fly in an optimized manner.Contrary to the conventional FFS which set the interfaces with equal distance of the order parameter,our approach determines the interfaces with equal transition probability which is shown to satisfy the optimization condition.This is done by firstly running long local trajectories starting from the current interface i to get the conditional probability distribution Pc(>i|i),and then determining i+1by equaling Pc(i+1|i)to a give value p0.With these optimized interfaces,FFS can be run in a much more efficient way.In addition,our approach can conveniently find the intermediate metastable states by monitoring some special long trajectories that neither end at the initial state nor reach the next interface,the number of which will increase sharply from zero if such metastable states are encountered.We apply our approach to a two-state model system and a two-dimensional lattice gas Ising model.Our approach is shown to be much more efficient than the conventional FFS method without losing accuracy,and it can also well reproduce the two-step nucleation scenario of the Ising model with easy identification of the intermediate metastable state.     

Reaction coordinates and transition pathways of rare events via forward flux sampling

A new approach is developed for identifying suitable reaction coordinates to describe the progression of rare events in complex systems. The method is based on the forward flux sampling (FFS) technique and standard least-square estimation (LSE) and it is denoted as FFS-LSE. The FFS algorithm generates trajectories for the transition between stable states as chains of partially connected paths, which can then be used to obtain "on-the-fly" estimates for the committor probability to the final region, p(B). These p(B) data are then used to screen a set of candidate collective properties for an optimal order parameter (i.e., reaction coordinate) that depends on a few relevant variables. LSE is used to find the coefficients of the proposed reaction coordinate model and an analysis of variance is used to determine the significant terms in the model. The method is demonstrated for several test systems, including the folding of a lattice protein. It is shown that a simple approximation to p(B) via a model linear on energy and number of native contacts is sufficient to describe the intrinsic dynamics of the protein system and to ensure an efficient sampling of pathways. In addition, since the p(B) surface found from the FFS-LSE approach leads to the identification of the transition state ensemble, mechanistic details of the dynamics of the system can be readily obtained during a single FFS-type simulation without the need to perform additional committor simulations.     

BAR-based optimum adaptive steered MD for configurational sampling

The equilibrium and nonequilibrium adaptive alchemical free energy simulation methods optimum Bennett's acceptance ratio and optimum crooks' equation (OCE), based on the statistically optimal bidirectional reweighting estimator named Bennett's Acceptance Ratio or Crooks' equation, perform initial sampling in the staging alchemical transformation and then determine the importance rank of different states via the time-derivative of the variance. The method is proven to give speedups compared with the equal time rule. In the current work, we extend the time derivative of variance guided adaptive sampling method to the configurational space, falling in the term of steered MD (SMD). The SMD approach biasing physically meaningful collective variable (CV) such as one dihedral or one distance to pulling the system from one conformational state to another. By minimizing the variance of the free energy differences along the pathway in an optimized way, a new type of adaptive SMD (ASMD) is introduced. As exhibits in the alchemical case, this adaptive sampling method outperforms the traditional equal-time SMD in nonequilibrium stratification. Also, the method gives much more efficient calculation of potential of mean force than the selection criterion-based ASMD scheme, which is proven to be more efficient than traditional SMD. The OCE workflow is periodicity-of-CV dependent while ASMD is not. The performance is demonstrated in a dihedral flipping case and two distance pulling cases, accounting for periodic and nonperiodic CVs, respectively. © 2019 Wiley Periodicals, Inc.     

Using the local elevation method to construct optimized umbrella sampling potentials: Calculation of the relative free energies and interconversion barriers of glucopyranose ring conformers in water

A method is proposed to combine the local elevation (LE) conformational searching and the umbrella sampling (US) conformational sampling approaches into a single local elevation umbrella sampling (LEUS) scheme for (explicit‐solvent) molecular dynamics (MD) simulations. In this approach, an initial (relatively short) LE build‐up (searching) phase is used to construct an optimized biasing potential within a subspace of conformationally relevant degrees of freedom, that is then used in a (comparatively longer) US sampling phase. This scheme dramatically enhances (in comparison with plain MD) the sampling power of MD simulations, taking advantage of the fact that the preoptimized biasing potential represents a reasonable approximation to the negative of the free energy surface in the considered conformational subspace. The method is applied to the calculation of the relative free energies of β‐D ‐glucopyranose ring conformers in water (within the GROMOS 45A4 force field). Different schemes to assign sampled conformational regions to distinct states are also compared. This approach, which bears some analogies with adaptive umbrella sampling and metadynamics (but within a very distinct implementation), is shown to be: (i) efficient (nearly all the computational effort is invested in the actual sampling phase rather than in searching and equilibration); (ii) robust (the method is only weakly sensitive to the details of the build‐up protocol, even for relatively short build‐up times); (iii) versatile (a LEUS biasing potential database could easily be preoptimized for small molecules and assembled on a fragment basis for larger ones). © 2009 Wiley Periodicals, Inc. J Comput Chem 2010     

Mechanistic elucidation of the yttrium(III)-catalysed intramolecular aminoalkene hydroamination: DFT favours a stepwise σ-insertive mechanism

A comprehensive computational mechanistic study regarding intramolecular hydroamination (HA) of aminoalkenes mediated by a recently reported class of highly active cyclopentadienyl-bis(oxazolinyl)borate {Cpo}Y(III) alkyl compounds is presented. Two distinct mechanistic pathways of catalytic HA mediated by rare earth and alkaline earth compounds have emerged over the years, describing amidoalkene → cycloamine conversion proceeding through a stepwise σ-insertive mechanism or a concerted non-insertive N-C/C-H bond forming pathway. Notably, both mechanisms account equally for reported distinct process features. Non-competitive kinetic demands revealed for the concerted amino proton transfer associated with N-C ring closure, which commences from a {Cpo(M)}Y(NHR)·(NH(2)R) substrate adduct and evolves through a six-centre TS structure, militates against a proton-triggered non-insertive pathway to promote HA for the rare earth catalyst at hand. A stepwise σ-insertive pathway, featuring rapid and reversible olefin insertion into the Y-N amido σ-bond, linked to a less facile and irreversible intramolecular Y-C azacycle tether aminolysis, is found to prevail energetically. The assessed effective barrier for turnover-limiting aminolysis matches the empirically determined Eyring parameter well and the computationally estimated primary KIE is close to the observed values. A recent computational study revealed a similar scenario for an analogous tris(oxazolinyl)borate {To(M)}Mg system. Valuable insights into the catalytic structure-reactivity relationships have been unveiled by a comparison of {Cpo(M)}Y(NHR)- and {To(M)}Mg(NHR)-catalysed hydroaminations.     

Synthesis and structures of selected triazapentadienate of Li, Mn, Fe, Co, Ni, Cu(I), and Cu(II) using 2,4-N,N'-disubstituted 1,3,5-triazapentadienate anions as ancillary ligands: [N(Ar)C(NMe2)NC(NMe2)N(R)](-) (Ar = Ph, 2,6-(i)Pr2-C6H3; R = H, SiMe3)

Addition reaction of ArN(SiMe 3)M (Ar = Ph or 2,6 - (i) Pr 2-C 6H 3 (Dipp); M = Li or Na) to 2 equivalents of alpha-hydrogen-free nitrile RCN (R = dimethylamido) gave the dimeric [M{N(Ar)C(NMe 2)NC(NMe 2)N(SiMe 3)}] 2 ( 1a, Ar = Ph, M = Li; 1b, Ar = Ph, M = Na; 1c, Ar = Dipp, M = Li). 1d was obtained by hydrolysis of 1c at ambient temperature. Treatment of a double ratio of 1a or 1b with anhydrous MCl 2 (M = Mn, Fe, Co) yielded the 1,3,5-triazapentadienato complexes [M{N(Ph)C(NMe 2)NC(NMe 2)N(SiMe 3)} 2] (M = Mn, 2; Fe, 3; Co, 4) and with NiCl 2.6H 2O gave [M{N(Ph)C(NMe 2)NC(NMe 2)N(H)} 2] (M = Ni, 5). Treatment of an equiv of 1c with anhydrous CuCl in situ and in air led to complexes [{N(Dipp)C(NMe 2)NC(NMe 2)N(SiMe 3)}CuPPh 3] 6 and [Cu{N(Dipp)C(NMe 2)NC(NMe 2)N(H)} 2] 7, respectively. 1c, 1d, and 2- 7 were characterized by X-ray crystallography and microanalysis. 1c, 1d, 5, and 6 were well characterized by (1)H, (13)C NMR, 1c by (7)Li, and 6 by (31)P NMR as well. The structural features of these complexes were described in detail.     

Oscillatory template exchange in polyoxometalate capsules: a ligand-triggered, redox-powered, chemically damped oscillation

The redox-controlled driven oscillatory template exchange between phosphate (P) and vanadate (V) anions enclosed in an {X(2)M(18)} cluster is reported. Extensive investigations using a range of techniques, including correlated ESI-MS, EPR, and UV-vis as a function of reaction time, showed that six complete oscillations interconverting the capsule species present in solution from {P(2)M(18)} to {V(2)M(18)} were possible, provided that a sufficient concentration of the TEA reducing agent was present in solution.     

Zinc-rich compounds of iron and cobalt: formation of [Fe2Zn(n)] (n = 2-4) and [Co2Zn3] cores

The reactions of heteroleptic GaCp*/CO containing transition metal complexes of iron and cobalt, namely [(CO)(3)M(μ(2)-GaCp*)(m)M(CO)(3)] (Cp* = pentamethylcyclopentadienyl; M = Fe, m = 3; M = Co, m = 2) and [Fe(CO)(4)(GaCp*)], with ZnMe(2) in toluene and the presence of a coordinating co-solvent were investigated. The reaction of the iron complex [Fe(CO)(4)(GaCp*)] with ZnMe(2) in presence of tetrahydrofurane (thf) leads to the dimeric compound [(CO)(4)Fe{μ(2)-Zn(thf)(2)}(2)Fe(CO)(4)] (1). Reaction of [(CO)(3)Fe(μ(2)-GaCp*(3))Fe(CO)(3)] with ZnMe(2) and stoichiometric amounts of thf leads to the formation of [(CO)(3)Fe{μ(2)-Zn(thf)(2)}(2)(μ(2)-ZnMe)(2)Fe(CO)(3)] (2) containing {Zn(thf)(2)} as well as ZnMe ligands. Using pyridine (py) instead of thf leads to [(CO)(3)Fe{μ(2)-Zn(py)(2)}(3)Fe(CO)(3)] (3) via replacement of all GaCp* ligands by three{Zn(py)(2)} groups. In contrast, reaction of [(CO)(3)Co(μ(2)-GaCp*)(2)Co(CO)(3)] with ZnMe(2) in the presence of py or thf leads in both cases to the formation of [(CO)(3)Co{μ(2)-ZnL(2)}(μ(2)-ZnCp*)(2)Co(CO)(3)] (L = py (4), thf (5)) via replacement of GaCp* with {Zn(L)(2)} units as well as Cp* transfer from the gallium to the zinc centre. All compounds were characterised by NMR spectroscopy, IR spectroscopy, single crystal X-ray diffraction and elemental analysis.     

Adaptive lambda square dynamics simulation: An efficient conformational sampling method for biomolecules

A novel, efficient sampling method for biomolecules is proposed. The partial multicanonical molecular dynamics (McMD) was recently developed as a method that improved generalized ensemble (GE) methods to focus sampling only on a part of a system (GEPS); however, it was not tested well. We found that partial McMD did not work well for polylysine decapeptide and gave significantly worse sampling efficiency than a conventional GE. Herein, we elucidate the fundamental reason for this and propose a novel GEPS, adaptive lambda square dynamics (ALSD), which can resolve the problem faced when using partial McMD. We demonstrate that ALSD greatly increases the sampling efficiency over a conventional GE. We believe that ALSD is an effective method and is applicable to the conformational sampling of larger and more complicated biomolecule systems. © 2013 Wiley Periodicals, Inc.     

Photoinduced magnetization with a high curie temperature and a large coercive field in a cyano-bridged cobalt-tungstate bimetallic assembly

A three-dimensional magnetic material [{CoII(pyrimidine)(H2O)}2{CoII(H2O)2}{WV(CN)8}2](pyrimidine)2. 2H2O is prepared. This compound exhibits a charge-transfer-induced spin transition with a large thermal hysteresis loop of 90 K. Irradiating with light causes the low-temperature phase to exhibit a spontaneous magnetization with a Curie temperature of 40 K and a magnetic hysteresis loop with a coercive field of 12 000 G, which is the highest value reported for a photomagnet. The observed photoinduced magnetization is due to the charge-transfer phase transition from the {CoIIhs(S = 3/2)}{CoIIIls(S = 0)}2-NC-{WIV(S = 0)}2 phase to the {CoIIhs(S = 3/2)}3-NC-{WV(S = 1/2)}2 phase by the irradiation.     

Efficient and precise solvation free energies via alchemical adiabatic molecular dynamics

A new molecular dynamics method for calculating free energies associated with transformations of the thermodynamic state or chemical composition of a system (also known as alchemical transformations) is presented. The new method extends the adiabatic dynamics approach recently introduced by Rosso et al. [J. Chem. Phys. 116, 4389 (2002)] and is based on the use of an additional degree of freedom, lambda, that is used as a switching parameter between the potential energy functions that characterize the two states. In the new method, the coupling parameter lambda is introduced as a fictitious dynamical variable in the Hamiltonian, and a system of switching functions is employed that leads to a barrier in the lambda free energy profile between the relevant thermodynamic end points. The presence of such a barrier, therefore, enhances sampling in the end point (lambda = 0 and lambda = 1) regions which are most important for computing relevant free energy differences. In order to ensure efficient barrier crossing, a high temperature T(lambda) is assigned to lambda and a fictitious mass m(lambda) is introduced as a means of creating an adiabatic separation between lambda and the rest of the system. Under these conditions, it is shown that the lambda free energy profile can be directly computed from the adiabatic probability distribution function of lambda without any postprocessing or unbiasing of the output data. The new method is illustrated on two model problems and in the calculation of the solvation free energy of amino acid side-chain analogs in TIP3P water. Comparisons to previous work using thermodynamic integration and free energy perturbation show that the new lambda adiabatic free energy dynamics method results in very precise free energy calculations using significantly shorter trajectories.     

On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition

In this paper, we investigated the spectrum of the operator \(L(\lambda )\) generated in Hilbert Space of vector-valued functions \(L_{2}(\mathbb {R}_{+},C_{2})\) by the system

$$\begin{aligned} iy_{1}^{'}(x,\lambda )+q_{1}(x)y_{2}(x,\lambda )&=\lambda y_{1}(x,\lambda )\\ -iy_{2}^{'}(x,\lambda )+q_{2}(x)y_{1}(x,\lambda )&=\lambda y_{2}(x,\lambda ),x\in \mathbb {R}_{+}:=(0,\infty ), \end{aligned}$$

and the integral boundary condition of the type

$$\begin{aligned} \int _{0}^{\infty }K(x,t)y(t,\lambda ){\mathrm {dt}}+\alpha y_{2}(0,\lambda )-\beta y_{1}(0,\lambda )=0 \end{aligned}$$

where \(\lambda \) is a complex parameter, \(q_{i},\,i=1,2\) are complex-valued functions and \(\alpha ,\beta \in \mathbb {C}\). K(x, t) is vector fuction such that \(K(x,t)=(K_{1}(x,t),K_{2}(x,t)),\,K_{i}(x,t)\in L_{1}(0,\infty )\cap L_{2}(0,\lambda ),\,i=1,2\). Under the condition

$$\begin{aligned} \left| q_{i}(x)\right| \le ce^{-\varepsilon \sqrt{x}},c>0,\varepsilon >0,i=1,2 \end{aligned}$$

we proved that \(L(\lambda )\) has a finite number of eigenvalues and spectral singularities with finite multiplicities.

     

A charge-transfer-induced spin transition in a discrete complex: the role of extrinsic factors in stabilizing three electronic isomeric forms of a cyanide-bridged co/fe cluster

A series of bimetallic, trigonal bipyramidal clusters of type {[Co(N-N)(2)](3)[Fe(CN)(6)](2)} are reported. The reaction of {Co(tmphen)(2)}(2+) with [Fe(CN)(6)](3)(-) in MeCN affords {[Co(tmphen)(2)](3)[Fe(CN)(6)](2)} (1). The cluster can exist in three different solid-state phases: a red crystalline phase, a blue solid phase obtained by exposure of the red crystals to moisture, and a red solid phase obtained by desolvation of the blue solid phase in vacuo. The properties of cluster 1 are extremely sensitive to both temperature and solvent content in each of these phases. Variable-temperature X-ray crystallography; (57)Fe Mossbauer, vibrational, and optical spectroscopies; and magnetochemical studies were used to study the three phases of 1 and related compounds, Na{[Co(tmphen)(2)](3)[Fe(CN)(6)](2)}(ClO(4))(2) (2), {[Co(bpy)(2)](3)[Fe(CN)(6)](2)}[Fe(CN)(6)](1/3) (3), and {[Ni(tmphen)(2)](3)[Fe(CN)(6)](2)} (4). The combined structural and spectroscopic investigation of 1-4 leads to the unambiguous conclusion that 1 can exist in different electronic isomeric forms, {Co(III)(2)Co(II)Fe(II)(2)} (1A), {Co(III)Co(II)(2)Fe(III)Fe(II)} (1B), and {Co(II)(3)Fe(III)(2)} (1C), and that it can undergo a charge-transfer-induced spin transition (CTIST). This is the first time that such a phenomenon has been observed for a Co/Fe molecule.     

Synthesis and structural and magnetic characterization of cobalt(II) phosphonate cage compounds

Reaction of cobalt salts with phosphonic acids in the presence of 6-chloro-2-hydroxypyridine as a co-ligand, normally in its deprotonated form, leads to a series of new polymetallic cobalt cages. The most common structural type is a {Co(14)} cage which resembles a fragment of cobalt hydroxide. Variation of the phosphonate present and the cobalt salt leads to {Co(6)}, {Co(8)}, {Co(10)}, {Co(11)}, {Co(12)}, {Co(13)}, and {Co(20)} cages, all of which have been characterized by X-ray crystallography. Magnetic studies of these cages show a general decline in the product chi(m)T with T, but for {Co(6)}, {Co(8)}, and {Co(12)} there are maxima at low temperature, which suggests nondiamagnetic ground states. Investigation of the dynamic behavior of the magnetization of these complexes shows that the octanuclear cage displays slow relaxation of magnetization.     

Synthesis and structures of monomeric magnesium, calcium, strontium, and barium amides involving the N,N-(2,4,6-trimethylphenyl)(trimethylsilyl)amido ligand

An extended family of aryl-substituted alkaline earth metal silylamides M{N(2,4,6-Me3C6H2)(SiMe3)}donor(n) was prepared using alkane elimination (Mg), salt elimination (Ca, Sr, Ba), and direct metalation (Sr, Ba). Three different donors, THF, TMEDA (TMEDA = N,N,N',N'-tetramethylethylenediamine), and PMDTA (PMDTA = N,N,N',N',N'-pentamethyldiethylenetriamine) were employed to study their influence on the coordination chemistry of the target compounds, producing monomeric species with the composition M{N(2,4,6-Me3C6H2)(SiMe3)}2(THF)2 (M = Mg, Ca, Sr, Ba), M{N(2,4,6-Me3C6H2)(SiMe3)}2TMEDA (M = Ca, Ba), and M{N(2,4,6-Me3C6H2)(SiMe3)}2PMDTA (M = Sr, Ba). For the heavier metal analogues, varying degrees of agostic interactions are completing the coordination sphere of the metals. Compounds were characterized using IR and NMR spectroscopy in addition to X-ray crystallography.     

Palladized aluminum as a novel substrate for electrosynthesis of polyaniline in sulfuric acid solutions

The electropolymerization of aniline on a palladized aluminum electrode (Pd/Al) by potentiodynamic as well as potentiostatic methods is described. The effect of the monomer concentration between 0.01 and 0.4 M on the polyaniline (PANI) formation and its growth on the Pd/Al electrode was investigated and a suitable concentration of 0.2 M is suggested. A similar study was carried out to investigate the effect of sulfuric acid concentration and 0.1 M sulfuric acid was chosen. A study on the influence of electropalladization time on the polymer formation and its growth suggested a convenient time of 40 s. The stability of the PANI film on the Pd/Al electrode was studied as function of the potential imposed on the electrode. For applied electrode potentials of 0.1–0.7 V, the first-order degradation rate constant, k, of PANI film varies between 1×10⁻⁶ and 2×10⁻⁵ s⁻¹, and a relatively low slope (i.e., 2.2) was obtained for the plot of log k versus E. The coatings were characterized by scanning electron microscopy (SEM), and cyclic voltammetric behavior of the PANI-deposited Pd/Al electrode is discussed. The electrocatalytic activity of the PANI-deposited Pd/Al electrode against para-benzoquinone/hydroquinone (Q/H₂Q) and redox systems were investigated and on the basis of of the corresponding cyclic voltammograms and the redox systems were identified as the reversible and quasi-reversible systems, respectively.     

Ln(terpy)]3+ (Ln = Sm, Gd) entity forms isolated magnetic chains with [W(CN)8]3-

The reaction between Ln(NO3)3*xH2O, Cs3[W(V)(CN)8]*H2O and 2,2':6',2'-terpyridine (terpy) leads to the original isomorphous cyano-bridged [Ln(III)(terpy)(DMF)4][W(V)(CN)8] *6H2O [Ln = Gd (1), Sm (2)] 1-D chains. The crystal structures of {Ln(III)W(V)} chains and consist of alternating {[W(CN)8]} and {[Ln(terpy)]} building blocks. The neighbouring 1-D chains are weakly linked through pi-pi stacking interactions of the aromatic rings leading to 2-D supramolecular layers. The layers are linked through hydrogen bonds between H2O molecules and terminal cyano ligands. Magnetic studies revealed a weak antiferromagnetic coupling (J = -2.3(2) K) within the {Gd(III)W(V)} chains in . The positive effective coupling constant J = +2.0(5) K between the total angular momentum of the Sm(III) centre and the spin of the W(v) ion is equivalent to an antiferromagnetic character of the spin coupling between both centres in the {Sm(III)W(V)} chains of 2. The magnetic measurements suggest that they display an isolated magnetic chain behaviour.     

A supramolecular heteropolyoxopalladate {Pd15} cluster host encapsulating a {Pd2} dinuclear guest: [Pd(II)2⊂{H7Pd(II)15O10(PO4)10}](9-)

A high-nuclearity polyoxopalladate compound, [Pd(II)(2)?{H(7)Pd(II)(15)O(10)(PO(4))(10)}](9-) {Pd(II)(17)}, comprising a {Pd(15)} host occupied by a {Pd(2)} guest and the parent pristine "empty" [H(7)Pd(II)(15)O(10)(P(V)O(4))(10)](13-) {Pd(15)} cluster have both been prepared and characterized by single-crystal X-ray crystallography, (31)P NMR, CSI-MS, and XPS. The encapsulated {Pd(2)} has a short Pd(II)-Pd(II) distance within the {Pd(15)} host. Solution studies indicate that the empty host and filled guest complex are in equilibrium with each other, and UV titrations revealed a binding constant of ca. 10(3) for the guest Pd(II) ions, with a binding stoichiometry of almost 2.     

Umbrella sampling for nonequilibrium processes

The authors introduce an algorithm for determining the steady-state probability distribution of an ergodic system arbitrarily far from equilibrium. By enforcing equal sampling of different regions of phase space, as in umbrella sampling simulations of systems at equilibrium, low probability regions are explored to a much greater extent than in physically weighted simulations. The algorithm can be used to accumulate joint statistics for an arbitrary number of order parameters for a system governed by any stochastic dynamics. They demonstrate the efficiency of the algorithm by applying it to a model of a genetic toggle switch which evolves irreversibly according to a continuous time Monte Carlo procedure.