chemistry across scales: from molecules to cells by sophia n yaliraki and mauricio barahona

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Sophia N Yaliraki and Mauricio BarahonaChemistry across scales: from molecules to cells

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Chemistry across scales: from moleculesto cells

BY SOPHIA N. YALIRAKI1,2,* AN D MAURICIO BARAHONA

1,3

1Institute for Mathematical Sciences, Imperial College London,London SW7 2PG, UK

2Department of Chemistry, Imperial College London, London SW7 2AY, UK 3Department of Bioengineering, Imperial College London, London SW7 2AZ, UK

Many important biological functions are strongly dependent on specific chemical

interactions. Modelling how the physicochemical molecular details emerge at much largerscales is an active area of research, currently pursued with a variety of methods. We describea series of theoretical and computational approaches that aim to derive bottom-updescriptions that capture the specificity that ensues from atomistic detail by extractingrelevant features at the different scales. The multiscale models integrate the descriptions atdifferent length and time scales by exploiting the idea of mechanical responses. Themethodologies bring together concepts and tools developed in seemingly unrelated areas of mathematics such as algebraic geometry, model reduction, structural graph theory andnon-convex optimization. We showcase the applicability of the framework with examplesfrom protein engineering and enzyme catalysis, protein assembly, and with the description

of lipid bilayers at different scales. Many challenges remain as it is clear that no singlemethodology will answer all questions in such multidimensional complex problems.

Keywords: multiscale dynamics of biomolecules; graph rigidity;

anisotropic interactions; model reduction; sum-of-squares decompositions

1. Multiscale dynamics of biomolecules: from chemical interactions todynamical responses

The realization that there is a molecular basis to many important biologicalfunctions, and conversely to loss of function leading to disease, has prompted ashift in the study of biological systems towards an emphasis on the chemicalinteractions that underpin biological and physiological processes. However,despite the wealth of generated data and simulations, a theoretical frameworkthat could lead to deeper mechanistic understanding, together with opportunitiesfor biological design and control, is still in its infancy. It is now clear that the studyof biological components in isolation is insufficient. At the same time, simulatingthe full spatial and temporal information of large number of molecules isimpractical and, even if possible, it would not necessarily increase our insight.

Phil. Trans. R. Soc. A (2007) 365, 2921–2934

doi:10.1098/rsta.2007.0015

Published online 13 September 2007

One contribution of 20 to a Triennial Issue ‘Chemistry and engineering’.

* Author and address for correspondence: Department of Chemistry, Imperial College London,London SW7 2AY, UK ([email protected]).

2921 This journal is q 2007 The Royal Society








Biological function is often the result of the action of tightly knit groups of molecules, hierarchically coupled in systems within systems into cellular andsupracellular arrangements. A major challenge for computational approaches tobiological simulation is to uncover how molecular interactions, which are specific,local and occur at the nano- and sub-nanometre scale, appear at larger scales where

physiological responses occur. Because these are highly complex systems in whichnonlinear and non-periodic interactions span orders of magnitude of time and lengthscales, naive attempts at compartmentalizing the dynamics often do not capture thekey underlying phenomena. In the post-genomic era, the challenge is now to developquantitative frameworks that can aid our understanding of biological behaviour, allthe way from the molecular level to the emergent properties at the organism level.To achieve this, an integrated multi-level approach is required where the essentialchemical components are identified and then integrated upwards taking intoaccount the metabolic network structures and the relevant physiological mechanics.Its defining characteristic is the integration of successful reduced descriptions across

multiple time and length scales. Our goal is to derive dynamical, bottom-updescriptions by sacrificing non-essential degrees of freedom, without losing thechemical specificity that ensues from the atomistic detail. Unlike continuum modelson the one hand, which ignore the discreteness of biological components, and staticnetworks on the other, where the nodes are either imposed or obtained from data, weseek to derive discrete, chemically based network models that, rather than beingstatic representations, can lead to dynamical models.

The development of low-resolution or coarse-grained approaches to biomolecularsystems is a very active area of research. For an excellent, recent review on thedifferent directions on methodology see Ayton et al. (2007). These approaches arealso relevant to lipid membranes (Brannigan et al. 2006), proteins (Tozzini 2005)and nucleic acids (Pyle & Widom 2006). Our focus here will be a particular set of methods motivated by recent advances in diverse areas of mathematics that arebased on nonlinearity, robustness and input–output optimality. These featureshave been largely ignored in thephysicochemical theoreticalcommunitybutmay beuseful in the characterization of biological processes. They lead to the use of a set of distinct mathematical tools (such as algebraic theory, model reduction, graphtheory and non-convex optimization) that have been developed in the area of engineering mathematics and theoretical computer science. Underpinning themathematical framework is the realization that often chemical interactions arecoupled with other scales through mechanical responses. For a wide range of

biological examples, from cells to tissues to organs that use mechanics in ahierarchical manner, see the recent review by Ingber (2006). Mechanical analogieshave been useful in coarse-graining chemical systems in the past, for example onharmoniclattices( Deutch&Silbey1971). Here, we show how starting fromdetailedatomic structures, we derive reduced units based on generic mechanical properties,such as shape and rigidity, that are well defined and accessible at all scales.

The dynamical behaviour of biomolecules provides a link between biologicalstructure and function. One of the outstanding successes of theoretical andcomputational chemistry has been the accurate representation of the dynamics of biomolecules through the development of carefully parametrized potentials to

describe chemicalinteractions (Brooks etal. 1983; Conrell etal. 1995).This molecular dynamics framework involves the numerical integration of nonlinear equations of motion of a very large number of degrees offreedom in contact with thermostats and

S. N. Yaliraki and M. Barahona 2922

Phil. Trans. R. Soc. A (2007)








barostats that ensure thermodynamic equilibrium (Hoover 1986). However, thereare computational limitations to the applicability of this approach to largebiomolecules and supramolecular assemblies, over the time scales that are relevantto biological function. Moreover, molecular dynamics simulations show that a largeproportion of the available degrees of freedom are usually ‘averaged out’ over time

and lead to non-observable dynamics. In this paper, we briefly introduce some ideasthat allow for a stratified, simplified description of biomolecular systems based onconstrained dynamics.

Firstly, we would like to obtain biologically relevant reduced descriptions of biomolecules that faithfully represent their dynamics. Owing to the intrinsiccomplexity of these systems, the mathematical tools invoked depart from thestandard linearization assumptions. We use graph theoretical concepts originallydeveloped for the analysis and design of mechanical structures, specificallyframeworks and rigidity theory (Connelly 1980; Connelly & Whiteley 1996), toeliminate degrees of freedom and simplify the molecular dynamics. This allows

for the biomolecules to be partitioned into quasi-rigid bodies and suggestsnaturally reduced representations for further study at larger scales. It alsohighlights the intimate link between structure and functional behaviour, as thepartitionings are derived from physicochemical interactions. We illustrate someof the results with applications to the mechanical modulation of enzyme activityand the emergence of supramolecular architecture at larger scales.

This dynamical partitioning leads to coarse-grained molecular models in whichthe concept of shape emerges naturally, since the resulting quasi-rigid bodies arealmost always anisotropic. The study of anisotropic interactions is a classic areain chemical physics (Onsager 1949; Berne & Pechukas 1972; Gay & Berne 1981;Perram et al. 1996) but there is surprisingly little work on the implication todynamics. We study how anisotropic reduced models can have importantimplications for structure and dynamics, and we concentrate on specific examplesinvolving lipid bilayer formation and Brownian motion of ellipsoidal particles.Another area of current research is the emergence of continuum properties fromassemblies of reduced models of biomolecules and how continuum propertiesdepend on molecular detail. We will highlight some of these issues with exampleswhere the physicochemical structure of lipids is shown to influence thespontaneous curvature of lipid bilayers and the activity of certain curvature-sensitive enzymes. This interdependence of ‘chemical composition–curvature–enzyme activity’ can lead to an intrinsic biophysical feedback mechanism that

regulates lipid biosynthesis (Attard et al. 2000). Throughout the paper, we willuse a series of model problems to exemplify the methodology which arenevertheless complex and relevant enough to merit research in their own right.

2. Towards dynamical coarse-graining of biomolecules: rigidityand model reduction

(a ) Graph rigidity for biomolecules

Our first goal is to identify rigid substructures in biomolecular structures. At the

smallest scale, this is translated into groups of atoms that move as ‘blocks’ and canthus be approximated as rigid bodies with frozen internal degrees of freedom.Although there is ongoing research into the use of linear methods (such as normal

2923Chemistry across scales









mode analysis and Gaussian network models; Bahar & Rader 2005; Tozzini 2005)to find relevant atom groupings, we have used a nonlinear, graph theoreticalapproach to identify such structures. The mathematical foundation follows fromrigorous results in graph and combinatorial rigidity theory (Connelly 1980) whichwere originally proved by Laman for planar graphs. These results can be extended

to a sub-class of graphs in three dimensions (bond-bending networks) that aregood descriptions of molecular structures (Jacobs 1998; Jacobs et al. 2001).

Starting from a full atom description that includes both covalent and non-covalent interactions, it is possible to derive a representation of a protein as a bond-bending network (a binary graph), where the nodes are the atoms and the edgesindicate distance constraints derived from the interatomic interactions. Conceptsfrom generic graph rigidity theory are then applied to identify flexible (under-constrained) and rigid (overconstrained or isostatic) regions (Jacobs 1998). Thesemechanical constraints can serve to identify, for example, which bonds can rotate;which bonds have correlated motions; and which segments of the protein behave as

rigid bodies. Continuous regions of consecutive rigid bonds define rigid clusters thatare interlinked by flexible bonds. This mechanical partitioning can be computedefficiently with FIRST, a computational tool for the analysis of proteins developedby Jacobs et al. (2001). FIRST implements a pebble game algorithm that scaleslinearly with the number of atoms in three-dimensional systems (Jacobs & Thorpe1995). An important feature of this graph theoretical approach is that it uses a full-atom description to obtain the distance constraints andhence derives theset of rigidclusters instead of assuming them a priori. In addition, the identification of available degrees offreedom and constrained regions is a combinatorial process andthus it is truly nonlinear. The advantage of this method is that it remains efficientcomputationally by avoiding costly energy minimizations, diagonalization of largematrices, or long molecular dynamics simulations.

(b ) The impact of rigidity on enzyme inhibition

This approach can be used to quantify the impact on rigidity of a givenintramolecular interaction or group of interactions. For example, the impact of ahydrogen bond can be assessed by removing the corresponding edges from thegraph and recalculating the rigidity of the protein. Because graph rigidity is anonlinear, non-local property, we expect such contributions to depend heavily onthe topology of the global network and the relative position of the interactions

and not only on the usual local rules based on energy. This application at theresidue level of description can be useful in elucidating the influence of residuepositions and substitutions, for example mutations, on the rigidity of the protein.The impact of each residue on the global rigidity depends both on its positionand on its contribution to the global hydrogen-bonding network. This structuralfeature can have implications for the functionality of proteins.

We have used this methodology to study natural and synthetic proteins of thefamily of the Bowman–Birk inhibitor (BBI), a class of naturally occurring serineprotease inhibitors that inhibit enzymes such as trypsin by binding to them(figure 1). These proteins share a conserved reactive site loop whose presence in a

conserved conformation is linked to their inhibitory activity. The most potentinhibitor known to date (found in the sunflower seed) consists of only this loop ina cyclic form (Brauer et al. 2002). In addition to naturally occurring instances,










there is considerable experimental effort devoted to the synthesis of BBI mimics,minimal in size but still functionally active, as well as the design of mutated oneswith maximal activity (McBride et al. 2002). Our results (Costa & Yaliraki 2006)show that this canonical loop is an independent rigid cluster in the naturalinhibitors. We have further used the technique to identify residues that play an

important role in the structural rigidity of the protein and relate their presenceto enzymatic activity. We find that the conserved elements among the naturaland synthetic peptides are those that contribute the most to rigidity, even if theyare located far from the active site. These results help to elucidate why certainmutations in the loop of the BBI produce peptides that fail to have the designedinhibitory activity.

The idea that the dynamical aspects of protein structure, and not onlythermodynamic measures such as binding energies, are important in catalysis isgaining ground (for a review see Benkovic & Hammes-Schiffer (2003)). Recentexperimental evidence showing that catalysis depends on the whole network of

residues, and not only on the active site ( Eisenmesser et al. 2005), highlights theneed for such new methodological analyses.

(c ) Dynamical coarse-graining from rigidity

The output from rigidity analysis can also be used to postulate coarse-grainedrepresentations for the dynamics of biomolecules based on block rigidity (Hemberget al. 2006). It is important to remark that the blocks are identified bottom-up, fromthe physicochemical interactions, without having to specify their number orstructure a priori. This course-graining approach leads to great simplification—starting typically from several thousand atomic coordinates it outputs represen-tations with tens of variables. These reduced models are then amenable to be usedeither within stochastic kinetics or rigid body dynamics frameworks. We havestudied such mechanical partitionings in individual proteins as well as insupramolecular assemblies of proteins involved in viral capsid formation.

Figure 2 shows an example of the partitioning of the protein adenylatekinase based on rigid blocks. This protein undergoes a large conformationalchange upon biding, where two extremal domains of the protein (the LIDdomain at the top and the AMP domain on the right) come together in a‘closing’ motion. Our rigidity analysis shows that most of the secondarystructure elements of the protein belong to different clusters. Moreover, it can

be seen that the LID and AMP domains form separate rigid clusters as wouldbe expected from the conformational change. The partitioning can be thentranslated into a reduced model in terms of rigid blocks which leads to areduced molecular dynamics. We have implemented this scheme computation-ally to evaluate the validity of these partitionings in a dynamical sense. Earlyresults indicate that the partitions based on rigidity are also meaningful forthe dynamics of the problem.

The reduction in the dimensionality of the problem opens the possibility of studying supramolecular assemblies of biomolecules and the process of self-assembly itself. Protein–protein interactions are central to cellular processes such

as signalling, genetic regulation and molecular recognition. Protein aggregationand self-assembly are also the biophysical processes underlying the formation of plaques and other aggregates in neurodegenerative diseases, such as










Huntington’s (Burke et al. 2003). Another important example of protein self-assembly is the formation of viral capsids. Each virus carries genetic materialthat can code for a particular coat protein. These proteins then assemble intohighly symmetric arrangements (icosahedral in many cases) of identical, non-symmetric proteins that surround and protect the viral genetic material. It isknown that the assembly pathways differ from virus to virus which indicates thatmolecular detail plays an important role in the process. We have developed abottom-up approach that starts with atomic descriptions of the individualproteins and performs rigidity analysis on the possible intermediate aggregates toestablish partitionings into rigid substructures (Hemberg et al. 2006). Thereduced descriptions are then used within a modified Gillespie algorithm to

sample the kinetics of self-assembly.Figure 3 shows some sample results of this approach, highlighting the fact

that the molecular fingerprint of the individual viral proteins translates into theappearance of distinct assembly pathways, both in structural intermediates andtime scales of formation. This influence of individual molecular characteristicson the emergent kinetics would be difficult to obtain from purely continuousmodels or from discrete phenomenological models that lack any atomistic detail.The observed differences are the result of distinctive intermediates that varyboth in association energies and dissociation propensities, which are affected bythe rigidity of the aggregates. Because it is a nonlinear, global property, the

rigidity of a particular unit may change dramatically as aggregation occurs,even though local neighbourhoods remain identical. As units associate, theglobal interaction graph changes leading to a redistribution of clusters during

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Figure 1. Rigidity effects on modification of enzyme activity (Costa & Yaliraki 2006). (a ) The peptideSFTI-1, one of the most potent known natural inhibitors from the sunflower seed: atomicrepresentation, residues and rigid cluster decomposition as obtained by FIRST (dark shades of blueindicate rigid bonds). (b) The acyclic peptide mimic (pdb 1GM2 acyclic) shows highly reducedinhibition power even though the active site (green residues) remains the same as above. Note thatchanges in the ‘non-active’side of the molecule lead nevertheless to loss of rigidity of the canonical loop.










system evolves in time, either stochastically as in the viral capsid assembly or

under molecular dynamics of rigid bodies as discussed above.If time is considered explicitly a priori, coarse-graining becomes more

involved. We have used exact approaches based on projector operator techniquesto provide solutions on linear model systems (Cubero & Yaliraki 2005a ).However, these methods have limited validity due to the memory of the kernel of the system, an issue of importance in reduced models of condensed phase systems(Cubero & Yaliraki 2005b). We have also explored an alternative approach wherewe use model reduction techniques to obtain a representation of the system overa finite time horizon. This is inspired by the input–output theory of modelreduction in the field of theoretical control. We have already shown that we can

recover good low-order approximations to the classic problem of the appearanceof dissipation (Barahona et al. 2002). We are currently exploring how thesemethods will deal with biomolecular systems.

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Figure 3. Molecular fingerprint in the assembly pathways of viral capsid formation (Hemberg et al.

2006). (a ) As the assembly progresses from the monomer to the full icosahedral capsid, the rigidity isre-calculated leading to different rigidity-based partitioning of the protein monomers involved. Proteinmonomers that form part of larger oligomers tend to be more rigid. Moreover, the change in overallrigidity is not continuous as specific oligomers have higher rigidity. This affects the assembly pathwaysfor particular viruses. (b) Assembly pathways of two viruses differ both in the structural intermediatesand in the time scales of formation. The assemblies of two viruses (with 1stm and 4sbv monomers)proceedthrough different pathways involving intermediates withdifferent numbers of monomers. In thecase of 1stm (i), dimers play a major role, whereas for 4sbv (ii), trimers and hexamers are the mainintermediates, as seen in the pathway summary and the transition map.










3. Molecular shape: the effect of anisotropy on dynamics

(a ) Anisotropic potentials for coarse-grained dynamics

We have described above a procedure based on mechanical and graph theoretical

concepts through which simplified structural descriptions can be derived ratherthan postulated. The induced partitionings bring to fore the importance of ‘shape’in biomolecules, i.e. the well-known fact that units in biology are anisotropic, withimportant consequences for system properties. However, anisotropic interactionsare important not only for structural or equilibrium properties but also forpacking and dynamical properties. For instance, surprising, and to date largelyignored, is the effect of anisotropy on Brownian motion. Anisotropic particles donot conform to the classical results and the coupling of translation and rotationleads to non-Gaussian behaviour, especially in confined environment as has beenrecently also experimentally observed (Han et al. 2006). The effects are predictedto be even stronger in the presence of external forces (Grima & Yaliraki 2007),

which suggests that they could have implications for the crowded cell.When describing the relevant mechanical properties of biomolecules through

coarse-graining, anisotropic potentials are a natural choice since they capture theintrinsic constraints introduced by the interactions. This is exemplified in figure 2,where we have represented the enclosing volumes for the atoms in the rigid clustersas ellipsoids. However, most standard approaches assume isotropic potentials (see,however, Liwo et al. 1997). General, rigorously defined and parametrizedanisotropic potentials are still lacking. Recently, we have shown that the classicanisotropic potential, the Gay–Berne ellipsoidal potential (Berne & Pechukas1972), can lead to artificial ordering forces because it does not represent volume

properly under all particle orientations. This observation could have implicationsfor the dynamical pathways of self-assembly where shape plays an important role.Inspired by the work of Perram and co-workers ( Perram et al. 1996), we haveprovided an alternative framework, by realizing that their geometric approachcould lead to the distance of minimal approach across the particle intercentrevector. We showed that this is a good approximation to the exact distance of closestapproach and developed an ellipsoidal potential which is generally applicable to anymixture of elliptic particles at any orientations and which scales correctly withdistance and captures the complexity of the well shapes associated with orientation(Paramonov & Yaliraki 2005). The potential is obtained using mathematical

techniques from algebraic theory and optimization.

(b ) Assembly of coarse-grained lipid bilayers

A key advantage of our approach is its computational applicability. Thederived potential is amenable to explicit, efficient dynamical simulation due tothe analytical form of the forces (Paramonov & Yaliraki 2005). We havedeveloped an in-house software that implements the full molecular dynamicsbased on this ellipsoidal potential (for details see Paramonov et al. (submitted))and have used it to simulate coarse-grained dynamics of lipid bilayer assembly.Some of the results of these coarse-grained simulations are shown in figure 4. The

obtained bilayers exhibit desired structural and dynamical features, i.e.regarding ordering, diffusion and fluctuation behaviours, which compare wellwith other models. We are currently pursuing a double direction. Firstly, we are










furthering the parametrization of the ellipsoidal potential to reflect specificmolecular properties. Secondly, we are studying averaged properties on largeensembles of coarse-grained lipids to relate them to (discretized) continuummechanical models. As seen in figure 4, quasi-continuum properties, such ascurvature and undulations, become more easily measurable in large ensembles.

(c ) Protein–membrane interactions and mechanical feedback

The goal of the preceding programme is to be able to introduce molecular detailinto higher-level models of the mechanics of the membrane. These descriptions are

becoming ever more important due to the growing experimental realization thatthe cell actively exploits and regulates the strong interdependence between thelipid composition of the membrane and its mechanical response. This is at the rootof several important functions, such as membrane trafficking, vesicle budding,growth, division and movement (McMahon & Gallop 2005).

We have started by studying the interaction of lipid bilayers with smallcompounds and with proteins. Small compounds are also at the heart of non-specific drug binding. Early results indicate that small compounds can penetratethe bilayer in some cases severely disrupting its local structure (Baciu et al. 2006).The interaction of proteins, specifically enzymes, with lipid bilayers is key to

several functions, in which there is an interplay between curvature-generating andcurvature-sensing proteins (Attard et al. 2000). We have studied in detail aparticular model in which some of the enzymes involved in lipid biosynthesis have

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Figure 4. Model lipid membranes at different scales. (a ) (i) Full atomistic description and (ii)harmonic element discretized model from Marcelli et al. (2005). (b) Two snapshots (i, ii) of dynamical simulations of bilayers obtained with the anisotropic coarse-grained model developed byParamonov & Yaliraki (2005).










been postulated to be curvature-sensitive, in the sense that their activity ismodulated by the stored elastic curvature energy of the lipid bilayer. Theseenzymes have amphipathic a-helices that bind more or less strongly to the bilayerdepending on its spontaneous curvature. The binding to the bilayer induces aconformational change that enhances the activity of the enzyme. This has the

effect of modifying the lipid composition of the membrane and provides abiophysical mechanism for feedback control between curvature and lipidcomposition. Our analysis to date has been biophysical and kinetic. There iswork in progress using the computational approach presented above to simulatebilayers with different lipids to estimate their spontaneous curvatures and to studytheir interaction with amphipathic ellipsoids.

(d ) Other applications

Since the developed potential is generic, it can be applied to any spherical oranisotropic coarse-grained unit of interest. For example, it can be combined with

the rigidity analysis as described above—the rigid bodies derived from atomistic-detailed protein structures could be modelled through anisotropic particles. Thismethodology could be used to explore the effects of anisotropy on the explicitdynamics of viral capsid assembly and protein aggregation, beyond the kineticapproach introduced above, by introducing a reduced dynamics with a level of structural detail derived from the molecular interactions.

It is also applicable to other problems where anisotropic interactions aredeterminant. This includes the classic packing problems for ellipsoids, which couldbe relevant for crowding in the cellular environment. When dealing with suchproblems, we have also exploited the fact that our potential is based on the

minimal distance between ellipsoids to recast the evaluation of configurations as ahigh-dimensional optimization problem. For the case of ellipsoids, this can besolved efficiently through a subset of semidefinite programmes ( Vandenberghe &Boyd 1996) which gives all the pairwise distances at once, together with proofsthat they have indeed been obtained. We have used this step in theimplementation of the molecular dynamics (Burke & Yaliraki 2006) and itcould also be used to perform quasi-static configurational searches. Importantly,this scheme is not restricted to ellipsoidal particles. A recently developedapproach, namely sum-of-squares (SOS) decomposition theory (Parrilo 2000),employs algebraic techniques to optimize nonlinear (polynomial) non-convex

problems. We are using this framework to extend this approach to non-convexbodies, thus increasing the fidelity of the coarse-grained descriptions.

4. Outlook

We have described here a series of coarse-graining directions which, starting fromatomistic descriptions to using mechanical concepts, aim to show how thetraditional chemistry of local interactions translates into global behaviours inbiological systems. We remark that the mechanical aspects are just the initialstep to be complemented through statistical mechanics in the future. Clearly,

many challenges remain and a variety of approaches should be pursued since forsuch nonlinear, multidimensional problems no method will outperform all othersin every situation.










This general approach of multiscale dynamics of biomolecules brings acomplementary approach to other areas of current interest in biological modellingwhich can enhance current research in molecular dynamics simulations forbiomolecules (driven by atomic potentials), bioinformatics (statistical analysis of static data) and systems biology (closed loop input–output biological modelling).

There are exciting times ahead as new methodologies are being developed to increasethe relevance and applicability of computational chemistry, mathematical modellingand systems engineering to biological andmedical systems. Ultimately, experimentalverification is important and a crucial partner in this endeavour.

We thank our research groups, especially Joao Costa, Martin Hemberg and Leonid Paramonov,who have contributed directly to the core of this work. We thank Kim Parker for fruitfuldiscussions. Funding from GSK, ONR (USA), EU-FP6, the Maths Institute at Imperial to S.N.Y.and from EPSRC and the Royal Society to M.B. is gratefully acknowledged.

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AUTHOR PROFILES

Sophia N. Yaliraki

Sophia Yaliraki graduated from Harvard University in 1992 and obtained a PhDin theoretical chemistry from MIT in 1997 under the supervision of Bob Silbey.She conducted postdoctoral work with Mark Ratner at Northwestern Universityin the area of molecular electronics. She joined Imperial College as a governorslecturer in 2000 and became a senior lecturer in 2004 and a reader in 2007. Since2006 she has been leader of the Multiscale Dynamics in Bio-systems Programmeat the newly formed Institute for Mathematical Sciences. Her current researchinterests are on the application of mathematical techniques from algebraicgeometry, optimization and graph theory to the development of reduced modelsthat can describe biological systems across different scales.

Mauricio Barahona

After completing his undergraduate studies at the Universidad Complutense inMadrid, Mauricio Barahona obtained a PhD in theoretical physics (dynamicalsystems) from MIT under the supervision of Steve Strogatz (Department of Mathematics). He was then a MEC postdoctoral fellow at Stanford University,

followed by his appointment as the Edison International Fellow at Caltech. He joined the newly formed Bioengineering Department at Imperial in 2001 as alecturer and became a reader in biomathematics in 2004. His current researchinterests include dynamics of interconnected nonlinear systems, specifically theconnections between graph theory and dynamics in genetic, metabolic andphysicochemical protein networks; theory of synchronization of coupledoscillators; algorithms for nonlinear data analysis; robust, multi-scale modelreduction of high-dimensional systems; and stochastic models of gene regulatorynetworks.








chemistry across scales: from molecules to cells by sophia n yaliraki and mauricio barahona

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