Interplay between multiple length and time scales in complex … · Interplay between multiple...

Interplay between multiple length and time scales incomplex chemical systems

BIMAN BAGCHI1 and CHARUSITA CHAKRAVARTY2,∗

1Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India.2Department of Chemistry, Indian Institute of Technology, Delhi, India.

∗e-mail: [email protected]

Processes in complex chemical systems, such as macromolecules, electrolytes, interfaces, micellesand enzymes, can span several orders of magnitude in length and time scales. The length and timescales of processes occurring over this broad time and space window are frequently coupled to giverise to the control necessary to ensure specificity and the uniqueness of the chemical phenomena.A combination of experimental, theoretical and computational techniques that can address a mul-tiplicity of length and time scales is required in order to understand and predict structure anddynamics in such complex systems. This review highlights recent experimental developments thatallow one to probe structure and dynamics at increasingly smaller length and time scales. The keytheoretical approaches and computational strategies for integrating information across time-scalesare discussed. The application of these ideas to understand phenomena in various areas, rangingfrom materials science to biology, is illustrated in the context of current developments in the areasof liquids and solvation, protein folding and aggregation and phase transitions, nucleation and self-assembly.

1. Introduction

An understanding of the diverse range of structuresand dynamical processes seen in chemical systemsis necessary in order to comprehend many phenom-ena in physics, chemistry and biology. Chemicaldiversity is generated by the interaction of well-defined components, electrons and nuclei, whosebehaviour is governed by well-understood forcesand equations of motion, corresponding to Coulom-bic forces and classical or quantum mechanics.Despite a century of progress in experimental,theoretical and computational physical chemistry,the application of physical principles to under-stand many aspects of such systems with reason-able predictive accuracy still poses a challenge.A major cause of difficulty is that the behaviourof a molecular system is determined by processesoperating on a multiplicity of length and timescales. Electrons moving on time scales of 10−18 s

(attoseconds) determine the interatomic interac-tions. Atomic motions are typically associated withlength and time scales of 10−10 m and 10−15 s,respectively; for example, time periods of mole-cular vibrations and rotations are of the order of10−13 s and 10−12 s, respectively. Many importantchemical reactions, like electron and proton trans-fer, occur on timescales of 10−9 to 10−12 s. Themesoscopic level of organization associated withmany soft matter systems, such as colloids, pro-teins and micelles, is associated with length/timescales of 10−6 m/10−9 s. Enzyme kinetics usuallyproceed with time constants of a few milliseconds,although the actual bond breaking/forming mayoccur in a few nanoseconds.

Tailoring the functional properties of materialson a macroscopic scale (1m/1 s) involves averagingover scales smaller by several orders of magni-tude in time and space. While statistical mechanicsdoes provide the theoretical framework for such

Keywords. Multiple time-scale methods; ultrafast spectroscopy; hasingle-molecule spectroscopy; energy landscapes;protein folding; protein aggregation; solvation; water; hydration; hydrophobic effect; nucleation; nanoscale self-assembly.

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68 BIMAN BAGCHI AND CHARUSITA CHAKRAVARTY

an averaging process, the actual implementation iscomplex because distinct experimental and compu-tational approaches are typically required to under-stand the behaviour of materials at different levelsof the hierarchy. For example, ab initio compu-tational methods for electronic structure focus onobtaining the electronic energies and wavefunctionsfor a given atomic arrangement [1,2]. In the macro-scopic limit, finite element approaches, based oncontinuum approximations that completely ignoreatomistic detail, are often used to model func-tional properties of materials, such as response tomechanical stress.

One of the traditional concerns of physical andtheoretical chemists has been and continues tobe improving the accuracy and range of toolsthat target specific levels in the scale hierarchy[3–7]. Provided the different levels in the hier-archy are only weakly coupled, such approachesare perfectly adequate for comparison of experi-ment and theory. Thus ab initio electronic struc-ture calculations are adequate for understandingultraviolet/visible spectra and classical moleculardynamics can be used to compute infra-red spectra.The current challenge is to understand phenomenathat are inherently multiscale, implying that twoor more levels in the hierarchy are strongly coupledbecause of which traditional theoretical or experi-mental tools are not applicable. This is the casewith some of the most interesting phenomena inmaterials science, chemistry and biology. A fami-liar example is that of electron transfer, where thequantum mechanical electron transfer process isintimately coupled to solvent reorganizations [8,9].An overall theoretical framework was provided byMarcus but the actual simulation of a realistic sys-tem still remains difficult. Enzyme catalysis pro-vides another example of such a strongly coupled,multiscale process [10]. The critical event involvingthe elementary chemical reaction of bond break-ing or formation of the substrate molecule takesplace locally and on a time scale of femtosec-onds. The steps that precede the catalytic reaction,such as the diffusion of the substrate to the activesite and the conformational reorganization of theenzyme necessary to lower the activation energy,are long on molecular time scales. The actual reac-tion rate constants for the catalytic process are ofthe order of milliseconds. In order to analyse andpredict enzymatic catalysis it is therefore necessaryto span electronic, atomic and mesoscopic lengthscales.

This review will focus on recent developmentsin physical chemistry which either enlarge thespectrum of length and time scales on which wecan probe chemical systems, or allow us to inte-grate information over many different levels of thelength/time scale hierarchy.

Our intention is to illustrate how a range ofinterdisciplinary problems can be addressed usingthese novel experimental and computational tech-niques in conjunction with the standard conceptsand tools of physical chemistry.

Section 2 provides an overview of two recenttechniques that can probe molecular systems atvery small spatial and temporal scales e.g., ultra-fast and single-molecule spectroscopies. These twotechniques represent fundamentally novel ways ofprobing the behaviour of chemical systems andhave resulted in new conceptual developmentsto interpret and design experiments in biology,chemistry and physics. Multi-scale computationaland theoretical approaches are reviewed in sec-tion 3. Sections 4, 5 and 6 illustrate how specificsystems and phenomena can be effectively under-stood using the experimental techniques and mul-tiple time-scale computational approaches outlinedin sections 2 and 3. Section 4 contains a discus-sion of some current areas of interest in liquid statedynamics and solvation. Protein folding and aggre-gation are regarded as among the most consequen-tial and difficult problems of structural biology andform the subject of section 5. Section 6 considersphase transition equilibria and kinetics, specificallywith reference to nanoscale self-assembly.

2. Experimental tools for probing lengthand time scales

Spectroscopy and scattering techniques are well-established tools for probing molecular structureand dynamics. Depending on the wavelength ofthe incident radiation, these techniques can beused to probe the system at atomic, electronicor nano-scales. Optical microscopes are increas-ingly used in conjunction with spectroscopy tostudy dynamical processes at the single moleculelevel. In this section, we do not attempt to reviewthese state-of-the-art techniques but focus on tworecent developments that have dramatically alteredour capability to probe the microscopic spatio-temporal correlations in molecular systems.

2.1 Ultrafast spectroscopy

G N Lewis once suggested that an appropriateunit of time for molecular processes would be the‘jiffy’, defined as the time taken by light to travel1 cm, or 33 picoseconds. Time scales shorter thanthis may be considered to be ‘ultrafast’. Betweenthe late 1960s and early 1980s, the technologydeveloped rapidly to enable the study of chemicalreactions in the nanosecond, and subsequently, inthe picosecond time domain. This period saw theemergence of a microscopic understanding of the

MULTIPLE LENGTH AND TIME SCALES IN COMPLEX CHEMICAL SYSTEMS 69

dynamics of vibrational energy and phase relax-ation and isomerization dynamics. At the sametime, it was realized that exploration of phenomenasuch as solvation dynamics, excited state electronand proton transfer reactions and dynamics atconical intersections required even shorter timeresolutions.

The late 1980s saw the birth of femtochem-istry with Ahmed Zewail reporting time-resolvedexperiments to detect the transition states involvedin chemical reactions, using what was termed asthe fastest camera in town [11,12]. In his NobelPrize lecture, Zewail described the crucial compo-nents of his camera as: (a) a femtosecond probelaser, determining the shutter speed, (b) a mecha-nism to record the image of the molecular systeme.g., using mass-spectrometry, diffraction or spec-troscopy (c) a femtosecond pump (initiating laser)which serves to set the zero of time and createa coherent wavepacket of molecular states whosemotion can be followed in real time. The designand interpretation of femtosecond experiments hasrequired considerable conceptual developments,including exploring the implications of molecularcoherence, development of a time-dependent for-mulation for spectroscopy and appropriate compu-tational tools to interpret femtosecond data. Theinitial ideas were developed and tested using sim-ple gas-phase reactions, such as dissociation of ICNand alkali halides. In the early experiments, it waspossible to observe transition states correspond-ing to unstable configurations of chemical systemslocated at the top of potential energy barriersseparating reactants and products. Today, the con-ceptual tools and the experimental techniques offemtochemistry are used to address questions inbiophysics, as will be discussed in section 4.

Moreover, it has now become possible to progressfrom femtosecond spectroscopy to attosecondspectroscopy in order to probe electronic motion[13,14].

2.2 Single-molecule spectroscopy

While ultrafast spectroscopy provides an astonish-ing array of time scales, its utility is sometimeslimited by the absence of spatial resolution. Thislacuna has been partly removed by single moleculespectroscopy (SMS) that probes the dynamics ofa single molecule in the condensed phase [15–17].SMS uses a combination of laser spectroscopy andoptical microscopy to achieve both temporal andspatial resolution, and is ideally suited to exploredynamics of protein folding, protein-DNA interac-tion and enzyme kinetics. The experimental designmust be such that concentrations are low enough toensure that only one molecule is in resonance in thevolume probed by the laser. Signal-to-noise ratios

must also be high enough that the signal from asingle chromophore can be detected despite thelarge number of solvent molecules and the intrinsicnoise of the measurement. With recent advancesin the field of microscopy, like confocal illumi-nation and two-photon microscopy, single mole-cule spectroscopy can completely remove ensembleaveraging and therefore allow the underlying dis-tributions to be determined, providing a powerfulprobe of static heterogeneity due to different nano-environments or conformational variations. Time-resolved variants of SMS spectroscopy can alsoaddress the dynamic heterogeneity associated withthe time-dependent variations in the behaviour ofa macromolecule, such as a protein. The review byMoerner discusses a number of milestones in SMS,including the possibility of creating single moleculesources of single photons on demand [15].

Single-molecule spectroscopy has proved to beparticularly useful in probing the structure andfunction of proteins [16,17]. For example, SMSexperiments offer the possibility of directly exami-ning the time-dependent folding processes of a sin-gle protein molecule. By encapsulating individualprotein molecules inside unilamellar protein vesi-cles tethered to surfaces, the time-dependent fold-ing and unfolding behaviour of adenylate kinasewas observed using fluorescence resonance energytransfer (FRET). In the case of enzymatic cataly-sis, the work of Xie and others demonstrates howstatic and dynamic heterogeneity can be experi-mentally studied in the case of redox reactionrates for flavoenzymes [17]. They demonstrate amolecular memory phenomenon, where enzymaticturnover depends on previous turnovers, due toslow conformational fluctuations.

3. Integrating information over multiplelength and time scales

3.1 Multiscale computational methodologies

Computation is now frequently projected as a toolfor discovery, complementing the traditional sci-entific methods of theory and experiment [18–22].This is particularly true for molecular systems inwhich computer simulation techniques play a cen-tral role. For such systems, computational tech-niques are valuable not just for understandingthe microscopic details, but also for exploringunexpected structures and emergent properties.As mentioned in the introduction, computationaltools for addressing separate length and timescales in the hierarchy are relatively well deve-loped. The current challenge is to develop effi-cient multiscale simulation methodologies whichcan integrate information from different levels of


Figure 1. Snapshots of a bilayer formed by a 1:1 mix-ture of cationic phospholipids stearoyl-decanoyl-ethylphos-phatidylcholine (C18:0/C10-EPC) and anionic phospho-lipid dioleoylphosphatidyl glycerol DOPG at 298 K (top)and an inverted hexagonal phase of a 1:1 mixture ofoleoyl-decanoyl-ethylphosphatidylcholine (C18:1/C10-EPC)and DOPG at 333 K (bottom). The coarse-grained modelused to simulate the lipid–water mixtures is discussed in[21]. (Reproduced with permission of American ChemicalSociety).

the hierarchy, particularly with a view to address-ing problems on the nano- or meso-scale. Fig-ures 1 and 2 address two such recent applications tointrinsically multi-scale problems. Figure 1 showsa picture of a lipid bilayer taken from [21]. Thestructure, dynamics and phase transitions of such

Figure 2. Atomistic configurations obtained with MDsimulations of propane nanojets emanating from a 6 nmdiameter nozzle, illustrating the development of instabilitiesand a double-cone pinch-off configuration (see [18]). Eachsmall blue sphere represents a propane molecule. (Copyright2005 National Academy of Sciences, USA).

a bilayer are strongly dependent on hydrophobicinteractions, and therefore on the solvent moleculesand their interaction with the macromolecularlipids. Some aspects of such a bilayer can, however,be treated at a coarse-grained level that ignoresatomistic details; for example, a lipid bilayer can betreated as a two-dimensional membrane characteri-zed by macroscopic variables, such as local mea-sures of elasticity. Figure 3 shows the formationof nanojets as propane gas escapes from a nozzle[18]. Formation and break-up of liquid jets are typi-cally modeled using continuum fluid dynamics butthe continuum approximations must be modifiedwhen the jet diameter approaches the nanoscale.Atomistic simulations provide a means to suggestas well as test such coarse-graining strategies.


Figure 3. Schematic illustration of the potential energylandscape of the condensed phases of an atomic system,taken from [31]. The potential energy will be a function ofthe position coordinates of all the atoms which is representedby the x-axis in this schematic diagram. The ordered lowenergy state corresponding to the perfect crystal is the globalminimum. The ideal glass state is an amorphous, packingminimum with energy very close to that of the crystal. In theliquid phase, the system makes frequent transitions betweenbasins of high energy local minima. Crystallisation requiresthat the system find its way from the manifold of high energyminima to the basin of the global minimum. Vitrificationcorresponds to trapping of the system in a metastable min-imum. Note the similarities with the free energy landscapefor protein folding shown in the next figure. (Figure repro-duced with permission of Nature Publishing Group).

The most tractable version of multiscaling isto use a sequential approach where simulationsof relatively small systems at lower levels of thehierarchy are used to generate input for higherlevels in the hierarchy [19,20]. A familiar exam-ple is the use of electronic structure calculationsto fit the parametric functional forms used torepresent interatomic interactions which can thenserve as input for atomistic simulations [23]. Thislevel of coarse-graining can be rigorously justifiedby the large difference in electronic and atomictime scales underlying the Born–Oppenheimerapproximation. Moving from the atomic to thenano-scale requires some thought regarding theappropriate type of information to be derived fromthe atomistic simulations in order to be used asinput for the mesoscale simulations. One approachis to construct effective potentials acting on themost important degrees of freedom, while averag-ing or integrating over the other atomistic degreesof freedom, such as those of the solvent molecules[19–21, 24–26]. Unlike in the case of interatomicpotentials, these effective potentials depend on thethermodynamic state point. Rigorous approachesto the coarse-graining attempt to preserve thepartition function but when this is not possible,the coarse-graining strategy is designed to ensure

that certain key thermodynamic quantities are pre-served. The snapshot of a lipid bilayer shown infigure 2 was taken from a simulation using such acoarse-grained potential. When studying mechani-cal properties of materials, atomistic simulationsmay be used to provide information on energies andstructures associated with crystalline defects ordislocations.

Unlike sequential methods, concurrent appro-aches to multiscaling attempt to simultaneouslysimulate the system at different levels of the hier-archy. Such methods are, in principle, more power-ful than sequential models because they make noassumptions about coarse-graining models and canbe applied when different levels of the hierarchyare strongly coupled. An example of a highly suc-cessful multiscale approach is ab initio or Car–Parrinello molecular dynamics that is appropriatewhen electronic structure and atomic scale motionsare strongly coupled, as in chemical reactions orin certain categories of phase transitions in whichthe nature of chemical bonding is altered [5,27].Interesting efforts are currently being made to for-mulate concurrent simulation methods that willlink the electronic, atomic and meso/macroscale bycombining ab initio methods, classical moleculardynamics and finite element methods, respectively.Examples of such approaches to study mechani-cal properties of materials, such as plasticity andfatigue, are given in [28–30].

3.2 Sampling energy landscapes

The energy landscape picture provides a conve-nient framework for understanding phase tran-sition thermodynamics and kinetics in complexchemical systems [31,32]. Potential energy land-scapes focus on the configurational energy as afunction of the 3N -dimensional position vector ofan interacting collection of N atoms. Assumingthat quantum effects are negligible on the atomicscale at temperatures of interest, the poten-tial energy surface contains all the informationnecessary to understand the collective properties.Energy landscape analyses attempt to characterisethe significant topographic features of the poten-tial energy landscape, such as number, locationand energies of minima and saddle points. Thesampling of the different regions of the energylandscape by a system in a given ensemble canbe simulated by generating a deterministic or astochastic trajectory, using molecular dynamics orMonte Carlo simulations. An alternative is to focuson the free energy of the system as a function ofcollective variables or order parameters. While thepotential energy landscape picture emerges natu-rally from a molecular simulation perspective, free


Figure 4. Schematic illustration of the potentialenergy surfaces involved in solvation dynamics showingthe water orientational motions along the solvation coordi-nate together with instantaneous polarization P (see [52]).In the inset we show the change in the potential energy alongthe intramolecular nuclear coordinate. As solvation proceedsthe energy of the solute comes down giving rise to a red shiftin the fluorescence spectrum. Note the instantaneous P , e.g.,P (�), on the two connected potentials (see [51,52]).

energy landscapes emerge naturally from a densityfunctional approach [4].

The potential energy landscape paradigm wasoriginally formulated in the context of supercooledliquids (see figure 4). In the solid phase, the systemis located in the basin of the global minimum whilein the liquid phase, it can make rapid transitionsbetween a manifold of metastable minima. Theproperties of the local minima and saddles of thisenergy landscape determine the thermodynamicand kinetic properties of the phases. For super-cooled liquids, the energy landscape picture hasyielded important insights into phenomenon suchas melting, supercooling and the glass transition.

The energy landscape view of liquids can beextended very succesfully to understand order-disorder transitions in complex fluids, such as theisotropic-nematic transition in liquids [33] and fold-ing of a solvated polypeptide chain into the nativeprotein structure [34]. In the case of protein fold-ing, it is more natural to think in terms of afree energy landscape by averaging over the sol-vent degrees of freedom, rather than the potentialenergy landscape. The native, folded configurationof the protein can be regarded as lying in the globalminimum of the system while the unfolded andpartially folded states lie in metastable free energyminimum. This energy landscape view of proteinfolding is discussed in greater detail in section 5.

The complexity of a chemical system can bedefined in terms of the structure of the energylandscape, including the multiplicity of minima,

the structure of the basins of minima and theirconnectivity. The time-scales governing the systemdynamics will be associated with bottlenecks incrossing between local minima. To understand thedynamics on a free energy landscape, it is con-venient to start with the simplest version of theproblem i.e., to view the chemical reaction betweentwo species A and B as a transition between twofree energy minima. The computational approachto determine the reaction rate for such a processfrom a molecular dynamics simulation is based onthe transition state theory and was formulated byBennett and Chandler in the late 1960s [3].

More recent algorithms to bridge time scales incomplex systems can be viewed as strategies toconstruct transition paths between minima sep-arated by free energy barriers of either enthalpicor entropic origin. If the key collective variableslikely to influence the dynamics are known, thenmethods such as umbrella sampling or paralleltempering can be used to construct the free energysurface. For example, in the case of solid-liquidphase transitions, the degree of crystallinity isclearly a suitable order parameter. Frequently,however, it is not possible to form an a priorijudgement of the critical collective variables ina complex system. The current interest centreson approaches to address precisely this class ofproblems, such as transition path sampling [35],hyperdynamics [36] and metadynamics [37]. Asthe number of accessible minima in the free energylandscape grows, corresponding to a multiplicity ofpossible reactants and products connected by ele-mentary reactions, it is often necessary to adopt amultiscale kinetic Monte Carlo (KMC) approach.For example, for surface growth phenomena, KMCsimulations combining kinetic energy barrier para-meters for relevant elemental processes from abinitio simulations can be used to predict spatio-temporal patterns associated with surface growth[38]. In the asymptotic limit of extreme complexityfound in many chemical, biological or genetic net-works, sophisticated algorithms may be requiredto merely deduce the relevant species and thereaction mechanism connecting them [39].

4. Liquids and solvation

Much of the chemistry takes place in the liquidstate with the solvent playing a critical rolein determining the overall thermodynamics andkinetics of various processes, including those asso-ciated with protein folding and nanoscale self-assembly discussed later. The dynamics of a liquidspans a huge range of time scales – from ultra-fast molecular level reorganisations to very slowdynamics associated with supercooling and the


glass transition. Theories of the liquid state areamong the more traditional domains of physicalchemistry, but many of the techniques and ideasare currently being employed to address problemsin complex fluids and soft matter. In this sec-tion, we consider the behaviour of a specific andubiquitous solvent (water), as well as some gen-eral features of solvation dynamics and the glasstransition.

Since water is the most common solvent forchemical and biological systems, understandingbulk water and its behaviour as a solvent is ofconsiderable interest [40,41]. Water has long beenknown to be an anomalous liquid in terms of itsdielectric, thermodynamic and kinetic properties.Computer simulations for water pose a challenge inthat the significant contribution of both dispersionand many-body polarizability contributions to theintermolecular interactions limit the accuracy ofboth classical and Car–Parrinello simulations. Thelight hydrogen atoms imply that quantum simu-lation methods are required for accuracy [42,43].Despite these limitations, classical Monte Carloand molecular dynamics simulations, in conjunc-tion with experiments, have resulted in uncoveringa number of unexpected features in the phase dia-gram of water, including new metastable phasesof ice and distinct polyamorphic (or glassy) forms[44,45]. An unexpected finding has been that theanomalous properties of water (such as the rise indensity on isobaric heating) are not uniquely asso-ciated with the tetrahedral, liquid-state network ofwater but may be seen in liquids with isotropicinterparticle interactions [46,47].

The hydrophobic interaction, most commonlymanifested as the segregation of oil and water, wasidentified by Kauzmann in 1959 as the driving forcein the collapse of a protein to its folded structure[48]. A molecular theory of hydrophobicity is essen-tial for understanding nanoscale organisation ofbiological structures and for developing appropri-ate simulation methodologies at both atomistic andmesoscopic scales (see figure 2). Recent work showsthat there is a strong length scale dependencein hydrophobic interactions [49]. Small hydropho-bic solutes induce some local order in the sur-rounding water but do not change the number ofhydrogen bonds in bulk water significantly. A largehydrophobic interface, however, actually disruptsthe hydrogen-bond network of water. The scalingof the free energy of hydration with the size of thehydrophobic solute therefore changes qualitativelyat a length scale of approximately 1 nm. In thecontext of biomolecular interfaces, the molecularpicture of hydrophobicity is further complicatedby the existence of surface chemical heterogeneity,since the protein surface will typically have bothhydrophobic and hydrophilic patches [50].

The dynamics of the solvation process, as des-cribed by the time-dependent response of the sol-vent to sudden changes in properties of the solute,was one of the first areas of chemistry wheretime-resolved techniques were successfully applied.Solvation dynamics has been extensively used toobtain important information on a wide variety ofsystems, from dipolar liquids like water and ace-tonitrile to complex systems like proteins, DNAmolecules, and micelles [51–56]. A schematic illus-tration of solvation as a dynamical phenomenonprobed by time domain laser experiments is shownin figure 4. One can experimentally construct asolvation time correlation function (STCF) thatmeasures the progress of solvation. This time scaleand also the exponentiality/non-exponentialityof the STCF serves to characterize the intrinsicdynamics of the complex solute-solvent system.From the perspective of statistical mechanics, onecan use linear response theory to describe thesolvation time correlation function in terms of anenergy-energy time correlation function which isthen related to the wavenumber and frequency-dependent dielectric function. The latter canbe obtained either from simulations or from ananalytical theory within certain approximations.The agreement between theory, simulations andexperiments has been tested for a diverse range ofsystems and proven to be very satisfactory. It isgenerally agreed that solvation dynamics in pureliquids measures dynamics over rather short dis-tances, although long range collective effects alsoplay a role. Bulk dipolar solvents, such as waterand acetonitrile, were found to exhibit ultrafastsolvation dynamics with sub-100 fs time constantsthough confinement of the solvents in reversemicelles can lead to a very slow solvation time scale.

Hydration dynamics at biomolecular interfacesshows a number of interesting features which haveinitiated a large amount of controversy and discus-sions [41]. At the protein surface, water moleculesare influenced both by the protein and the bulkwater. It is now believed that water molecules forma two dimensional network structure around theprotein molecule, with an associated modificationof water structure up to about 2–4 molecular layersof water. Several recent studies have explored watersolvation and orientational dynamics in the majorand minor grooves of DNA [56]. The slow dynamicsin the minor groove was interpreted partly as dueto nano-confinement and partly due to interactionwith the polar base atoms in the minor groove.

As a liquid is supercooled, it can either undergoa nucleation process to form the stable solid orundergo a glass transition. The glass transitionis experimentally defined as the temperature atwhich the viscosity of a liquid reaches 1013 poise.The key features of the glass transition are the


simultaneous appearance of slow non-exponentialrelaxation and non-Arrhenius temperature depen-dence [57,58]. Understanding the nature of theglass transition is an important issue in condensedmatter theory and statistical mechanics. Giventhe technological importance of glass in a vari-ety of contexts (e.g., metallic glass and amorphoussilicon), the glass transition has been subject ofconsiderable experimental and theoretical work.A number of different theoretical approaches havebeen developed to understand the glass transition.Energy landscape approaches have been employedto understand the connection between thermody-namics and dynamics in the stable and metastablesupercooled liquid state [59]. Mode coupling theorycan successfully explain the initial growth of relax-ation time on approaching the glass transition butis known to falter at still lower temperatures whereviscosity becomes larger than approximately 100poise [60,61]. An important issue is the dynamiccorrelation length which characterizes coopera-tively rearranging regions (CRR) of the liquid thatemerges as the glass transition is approached. Bothexperiments and theory suggest a size of a few nmfor these systems. It has been suggested that themorphology of these CRRs will show a crossoverfrom being fractal, close to the mode-coupling tem-perature to compact structures, close to the glasstransition [62].

5. Protein folding and aggregation

Understanding the physical principles that governthe folding of an extended polypeptide chaininto the characteristic three-dimensional tertiarystructure or ‘native’ state constitutes one ofthe fundamental problems of natural science.Since Anfinsen’s statement of the thermodynamichypothesis more than half-a-century ago, the pro-tein folding problem has been an area whereexperiment, theory and computations have inter-acted very effectively [63–66]. The thermody-namic hypothesis states that ‘sequence dictatesstructure’, i.e., the native state represents the mini-mum free energy structure for the polypeptidechain in its physiological millieu. Protein fold-ing times under physiological conditions are ofthe order of nanoseconds to milliseconds. TheLevinthal paradox encapsulates the difficulty ofreconciling the enormous number of conforma-tional structures with relatively short protein fold-ing time scales. The diversity of folding timesand the complexity of the intracellular millieu,however, implies that sequential folding pathwaysguided solely by thermodynamic stability are likelyto exist only in a small category of fast-foldingproteins. The challenge of understanding protein

Figure 5. A schematic free energy landscape for proteinfolding. The surface is derived from a computer simulationof the folding of a highly simplified model of acylphophatase,with constraints derived from experimental data from muta-tional studies. The yellow spheres in this ensemble representthe three ‘key residues’ in the structure; when these residueshave formed their native-like contacts the overall topologyof the native fold is established. The native state lies at thebottom of the free energy funnel and possible folding path-ways are shown with arrows (figure reproduced with permis-sion of Nature Publishing Group).

folding is complicated by the fact that computersimulations of the energetics and dynamics ofa polypeptide chain in an aqueous medium arelimited in terms of simulation size, length oftime for which the protein dynamics can be fol-lowed and the accuracy with which the under-lying interatomic interactions can be modeled.Despite the complexity of the problem, a com-bination of experimental techniques (NMR, opti-cal spectroscopies, crystallography), multi-scaleand coarse-grained simulations and insights fromthe statistical mechanics of liquids and polymershave led to the development of an energy landscapeparadigm as a conceptual framework for under-standing protein folding. Figure 5 shows the freeenergy landscape of a typical protein, constructedusing simulation and NMR data. The free energyis mapped out as a function of two key collectivevariables: the number of native interactions andthe number of residue contacts. The various pro-tein conformations can be seen to occupy differentparts of a funnel in the free energy landscape whichterminates at the native structure. The unfolded,random coil configurations of the protein occupythe mouth of the channel. Intermediate statesformed by hydrophobic collapse and secondarystructure formation lie along the funnel and definethe folding pathways. The shortest time-scales inprotein folding are associated with the formation


of secondary structures, such as helices while thelong time scales are associated with the formationof tertiary structures involving contacts betweenamino acid residues well-separated along the pri-mary amino acid sequence. The landscape picturecan accomodate multiple pathways and transitionstate ensembles. When entropic and enthalpic con-tributions to the free energy compensate, theremay be no significant barriers to folding, leadingto very fast, ‘downhill’ folding.

The existence of complex protein folding path-ways implies that in the cellular enviroment, theintermediate structures are available for interactionwith other intracellular components. Some of thesemay be molecular chaperones which assist the fold-ing process whereas others may lead to misfolding.Protein aggregation, specially the formation offibrillar aggregates, leads to severe pathologies.The nucleation processes which lead to foldingversus aggregation are consequently an activearea of research. Another active area of researchwhere many of the ideas of protein folding andaggregation can be applied is in biomimetic strate-gies for creating non-biological polymers that self-assemble to form stable, unique structures indifferent solvents [67–69].

6. Phase transitions, nucleation andself-assembly

The equilibrium phase diagram of a system sum-marises the range of thermodynamically stablebulk structures that can be generated from a set ofatomic or molecular constituents. The generationof phase diagrams for a range of systems is todaypossible using a variety of simulation techniques,where the predictive accuracy of the simulationsdepends largely on the accuracy with which theunderlying interactions can be modeled [3]. Thekinetics of phase transformations, in contrast tothe thermodynamics, is currently an active area ofwork. First-order transitions are assumed to formby a nucleation process where an embryonic sta-ble phase forms within the metastable phase, inthe absence (homogeneous) or presence (heteroge-neous) of surfaces or impurities. Classical nucle-ation theory assumes that free energy required tocreate a nucleus of the stable phase in a metastablematrix depends on two competing terms: a sur-face term that represents the free energy requiredto form an interface and a volume term that rep-resents the free energy gain proportional to thedifference in chemical potential between the stableand metastable phases [57]. Nuclei greater than acritical size will grow spontaneously. The forma-tion of a nucleus of critical size is thus the ratedetermining step and takes place on time scales

that are very long compared to molecular timescales. Since a complex system can form multiplephases, crystallization from a melt can frequentlyresult in the formation of metastable phases ratherthan stable phases.

Appropriate simulation methodologies to bridgetime scales for nucleation in molecular simulationsare currently an active area of work. Computa-tional studies of nucleation have largely focusedon the condensation and crystallisation processesin relatively simple one-component systems thatallow one to address a number of interestingtheoretical possibilities, including the effect ofmetastable phases on crystallization and conden-sation in the vicinity of the liquid-gas spinodal[70–74]. One finds that at large supersaturation(near the spinodal), the mechanism of nucleationundergoes a remarkable change from classical sin-gle particle addition to a collective mode, spreadover the entire system (figure 6). The critical step isnow the coalescence of clusters of intermediate sizeon a flat free energy surface (figure 6) [74]. Thesestudies have interesting implications for nucleationin more complex systems, e.g., crystallization ofproteins or obtaining stable polymorphs of phar-maceuticals.

Phase transitions under equilibrium and non-equilibrium conditions are of special interest inthe context of nanoscale self-assembly of orderedstructures [75–78]. Advances in nanoparticle syn-thesis have made it possible to synthesize nanopar-ticles in a large variety of shapes and sizes. Thesenanoparticles can self-assemble to form a remark-able range of ordered structures (see figure 7 for anillustration of a miniscule sample of possible struc-tures). Under a given set of macroscopic conditionswhich may not necessarily be equilibrium condi-tions, an ordered self-assembled structure is formedas a consequence of the interplay between the vari-ous contributions to interparticle interactions, suchas dispersion, short-range repulsion, electrostatic,bonding and solvophobic interactions. Understand-ing the organisational principles of self-assemblyand developing the appropriate predictive tools iscurrently a major challenge.

7. Concluding remarks

This article attempts to provide an overview ofthe interdisciplinary array of theoretical, compu-tational and experimental approaches that arecurrently being developed in order to understandthe interplay between multiple length and timescales that determine structure and dynamics inmany chemical systems. The choice of illustra-tive examples has been inevitably biased by ourown research interests but we hope that it is


Figure 6. (a) The 3-dimensional free energy surface (upper panel) and corresponding 2-dimensional contour plot (lowerpanel) computed in grand canonical (μVT) ensemble at supersaturation ≈2.4 (near gas-liquid spinodal). (b) Snapshotsof the system at four different supersaturation (S = P/Pc, where P is the pressure of the system and Pc is the same atthe gas-liquid coexistence.): They show all the liquid-like particles of the system at the top of the barrier. For highersupersaturation, we find multiple large clusters are forming around the critical cluster and growth of the liquid phasebecomes spread over the whole system rather than the single ‘critical cluster’ (see [74]).

Figure 7. TEM images of self-assembled structures and proposed unit cells of binary superlattices (a) and (b) structureformed from LaF3 triangular nanoplates (9.0 nm side) and 5.0 nm Au nanoparticles; (c) structure formed from LaF3 trian-gular nanoplates and 6.2 nm PbSe nanocrystals. Details are available in [76]. (Figure reproduced with permission of NaturePublishing Group).

sufficient to illustrate that currently available tech-niques can address a number of interesting prob-lems in the areas of materials science, chemical

biology and condensed matter physics. Constraintsof space in a volume of this type have meantthat we have not been able to review experimental


and theoretical research on areas of related inter-est in India. There are currently several excel-lent groups working in the area of spectroscopy,scattering, electronic structure methods, statisti-cal mechanics, liquid state theory, atomistic simu-lations and continuum methods. These groups aretypically dispersed over chemistry, physics, chemi-cal engineering and materials science departments.We hope that this article will illustrate that thereis considerable scope as well as need for ‘multi-displinary’ as well as ‘multiscale’ approaches tounderstand a wide range of problems.

Acknowledgements

This work was supported in parts by grants fromDST and CSIR, India. We thank Sangeeta Saini,Abir Ganguly, Shadrack Jabes and Biman Jana forhelp in preparing the manuscript.

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