Biomolecules and their function(s).

 Computational Structural Biology (CSB) is the scientific domain concerned with the development of algorithms and software to understand and predict the structure and function of biological macromolecules. This research field is inherently multi-disciplinary. On the experimental side, biology and medicine provide the objects studied, while biophysics and bioinformatics supply experimental data, which are of two main kinds. On the one hand, genome sequencing projects give supply protein sequences, and ~200 millions of sequences have been archived in UniProtKB/TrEMBL – which collects the protein sequences yielded by genome sequencing projects. On the other hand, structure determination experiments (notably X-ray crystallography, nuclear magnetic resonance, and cryo-electron microscopy) give access to geometric models of molecules – atomic coordinates. Alas, only ~150,000 structures have been solved and deposited in the Protein Data Bank (PDB), a number to be compared against the $\sim {10}^8$ sequences found in UniProtKB/TrEMBL. With one structure for ~1000 sequences, we hardly know anything about biological functions at the atomic/structural level. Complementing experiments, physical chemistry/chemical physics supply the required models (energies, thermodynamics, etc). More specifically, let us recall that proteins with $n$ atoms has $d=3n$ Cartesian coordinates, and fixing these (up to rigid motions) defines a conformation. As conveyed by the iconic lock-and-key metaphor for interacting molecules, Biology is based on the interactions stable conformations make with each other. Turning these intuitive notions into quantitative ones requires delving into statistical physics, as macroscopic properties are average properties computed over ensembles of conformations. Developing effective algorithms to perform accurate simulations is especially challenging for two main reasons. The first one is the high dimension of conformational spaces – see $d=3n$ above, typically several tens of thousands, and the non linearity of the energy functionals used. The second one is the multiscale nature of the phenomena studied: with biologically relevant time scales beyond the millisecond, and atomic vibrations periods of the order of femto-seconds, simulating such phenomena typically requires $\gg {10}^{12}$ conformations/frames, a (brute) tour de force rarely achieved 37.

Computational Structural Biology: three main challenges.

 The first challenge, sequence-to-structure prediction, aims to infer the possible structure(s) of a protein from its amino acid sequence. While recent progress has been made recently using in particular deep learning techniques 36, the models obtained so far are static and coarse-grained.

The second one is protein function prediction. Given a protein with known structure, i.e., 3D coordinates, the goal is to predict the partners of this protein, in terms of stability and specificity. This understanding is fundamental to biology and medicine, as illustrated by the example of the SARS-CoV-2 virus responsible of the Covid19 pandemic. To infect a host, the virus first fuses its envelope with the membrane of a target cell, and then injects its genetic material into that cell. Fusion is achieved by a so-called class I fusion protein, also found in other viruses (influenza, SARS-CoV-1, HIV, etc). The fusion process is a highly dynamic process involving large amplitude conformational changes of the molecules. It is poorly understood, which hinders our ability to design therapeutics to block it.

Show image description

The spike of SARS-CoV-2 is responsible for the infection. Molecules can be engineered to compete with the receptor of the spike, and block infection.

Finally, the third one, large assembly reconstruction, aims at solving (coarse-grain) structures of molecular machines involving tens or even hundreds of subunits. This research vein was promoted about 15 years back by the work on the nuclear pore complex 25. It is often referred to as reconstruction by data integration, as it necessitates to combine coarse-grain models (notably from cryo-electron microscopy (cryo-EM) and native mass spectrometry) with atomic models of subunits obtained from X ray crystallography. Fitting the latter into the former requires exploring the conformation space of subunits, whence the importance of protein dynamics.

As an illustration of these three challenges, consider the problem of designing proteins blocking the entry of SARS-CoV-2 into our cells (Fig. 1). The first challenge is illustrated by the problem of predicting the structure of a blocker protein from its sequence of amino-acids – a tractable problem here since the mini proteins used only comprise of the order of 50 amino-acids (Fig. 1(A), 28). The second challenge is illustrated by the calculation of the binding modes and the binding affinity of the designed proteins for the RBD of SARS-CoV-2 (Fig. 1(B)). Finally, the last challenge is illustrated by the problem of solving structures of the virus with a cell, to understand how many spikes are involved in the fusion mechanism leading to infection. In 28, the promising designs suggested by modeling have been assessed by an array of wet lab experiments (affinity measurements, circular dichroism for thermal stability assessment, structure resolution by cryo-EM). The hyperstable minibinders identified provide starting points for SARS-CoV-2 therapeutics 28. We note in passing that this is truly remarkable work, yet, the designed proteins stem from a template (the bottom helix from ACE2), and are rather small.

Show image description

Prosaically, the energy landscape of a molecular system is the set of valleys and passes connecting them, corresponding to stable states and transitions between them.

Protein dynamics: core CS - maths challenges.

To present challenges in structural modeling, let us recall the following ingredients (Fig. 2). First, a molecular model with $n$ atoms is parameterized over a conformational space $𝒳$ of dimension $d=3n$ in Cartesian coordinates, or $d=3n-6$ in internal coordinate–upon removing rigid motions, also called degree of freedom (d.o.f.). Second, recall that the potential energy landscape (PEL) is the mapping $V(·)$ from ${ℝ}^d$ to $ℝ$ providing a potential energy for each conformation 38, 35. Example potential energies (PE) are CHARMM, AMBER, MARTINI, etc. Such PE belong to the realm of molecular mechanics, and implement atomic or coarse-grain models. They may embark a solvent model, either explicit or implicit. Their definition requires a significant number of parameters (up to $\sim 1,000$), fitted to reproduce physico-chemical properties of (bio-)molecules 39.

These PE are usually considered good enough to study non covalent interactions – our focus, even tough they do not cover the modification of chemical bonds. In any case, we take such a function for granted 1.

The PEL codes all structural, thermodynamic, and kinetic properties, which can be obtained by averaging properties of conformations over so-called thermodynamic ensembles. The structure of a macromolecular system requires the characterization of active conformations and important intermediates in functional pathways involving significant basins. In assigning occupation probabilities to these conformations by integrating Boltzmann's distribution, one treats thermodynamics. Finally, transitions between the states, modeled, say, by a master equation (a continuous-time Markov process), correspond to kinetics. Classical simulation methods based on molecular dynamics (MD) and Monte Carlo sampling (MC) are developed in the lineage of the seminal work by the 2013 recipients of the Nobel prize in chemistry (Karplus, Levitt, Warshel), which was awarded “for the development of multiscale models for complex chemical systems”. However, except for highly specialized cases where massive calculations have been used 37, neither MD nor MC give access to the aforementioned time scales. In fact, the main limitation of such methods is that they treat structural, thermodynamic and kinetic aspects at once 31. The absence of specific insights on these three complementary pieces of the puzzle makes it impossible to optimize simulation methods, and results in general in the inability to obtain converged simulations on biologically relevant time-scales.

The hardness of structural modeling owes to three intertwined reasons.

First, PELs of biomolecules usually exhibit a number of critical points exponential in the dimension 26; fortunately, they enjoy a multi-scale structure 29. Intuitively, the significant local minima/basins are those which are deep or isolated/wide, two notions which are mathematically qualified by the concepts of persistence and prominence. Mathematically, problems are plagued with the curse of dimensionality and measure concentration phenomena. Second, biomolecular processes are inherently multi-scale, with motions spanning $\sim$ 15 and $\sim$ 4 orders of magnitude in time and amplitude respectively 24. Developing methods able to exploit this multi-scale structure has remained elusive. Third, macroscopic properties of biomolecules, i.e., observables, are average properties computed over ensembles of conformations, which calls for a multi-scale statistical treatment both of thermodynamics and kinetics.

Validating models.

A natural and critical question naturally concerns the validation of models proposed in structural bioinformatics. For all three types of questions of interest (structures, thermodynamics, kinetics), there exist experiments to which the models must be confronted – when the experiments can be conducted.

For structures, the models proposed can readily be compared against experimental results stemming from X ray crystallography, NMR, or cryo electron microscopy. For thermodynamics, which we illustrate here with binding affinities, predictions can be compared against measurements provided by calorimetry or surface plasmon resonance. Lastly, kinetic predictions can also be assessed by various experiments such as binding affinity measurements (for the prediction of $K_{on}$ and $K_{off}$), or fluorescence based methods (for kinetics of folding).

3.1 Modeling the dynamics of proteins

Keywords: Molecular conformations, conformational exploration, energy landscapes, thermodynamics, kinetics.

As noticed while discussing Protein dynamics: core CS - maths challenges, the integrated nature of simulation methods such as MD or MC is such that these methods do not in general give access to biologically relevant time scales. The framework of energy landscapes 38, 35 (Fig. 2) is much more modular, yet, large biomolecular systems remain out of reach.

To make a definitive step towards solving the prediction of protein dynamics, we will serialize the discovery and the exploitation of a PEL 4, 16, 3. Ideas and concepts from computational geometry/geometric motion planning, machine learning, probabilistic algorithms, and numerical probability will be used to develop two classes of probabilistic algorithms. The first deals with algorithms to discover/sketch PELs, i.e., enumerate all significant (persistent or prominent) local minima and their connections across saddles, a difficult task since the number of all local minima/critical points is generally exponential in the dimension. To this end, we will develop a hierarchical data structure coding PELs as well as multi-scale proposals to explore molecular conformations. (NB: in Monte Carlo methods, a proposal generates a new conformation from an existing one.) The second focuses on methods to exploit/sample PELs, i.e., compute so-called densities of states, from which all thermodynamic quantities are given by standard relations 2734. This is a hard problem akin to high-dimensional numerical integration. To solve this problem, we will develop a learning based strategy for the Wang-Landau algorithm 33–an adaptive Monte Carlo Markov Chain (MCMC) algorithm, as well as a generalization of multi-phase Monte Carlo methods for convex/polytope volume calculations 32, 30, for non convex strata of PELs.

3.2 Algorithmic foundations: geometry, optimization, machine learning

Keywords: Geometry, optimization, machine learning, randomized algorithms, sampling, optimization.

As discussed in the previous Section, the study of PEL and protein dynamics raises difficult algorithmic / mathematical questions. As an illustration, one may consider our recent work on the comparison of high dimensional distribution 7, statistical tests / two-sample tests 8, 13, the comparison of clustering 9, the complexity study of graph inference problems for low-resolution reconstruction of assemblies 12, the analysis of partition (or clustering) stability in large networks, the complexity of the representation of simplicial complexes 2. Making progress on such questions is fundamental to advance the state-of-the art on protein dynamics.

We will continue to work on such questions, motivated by CSB / theoretical biophysics, both in the continuous (geometric) and discrete settings. The developments will be based on a combination of ideas and concepts from computational geometry, machine learning (notably on non linear dimensionality reduction, the reconstruction of cell complexes, and sampling methods), graph algorithms, probabilistic algorithms, optimization, numerical probability, and also biophysics.

3.3 Software: the Structural Bioinformatics Library

Keywords: Scientific software, generic programming, molecular modeling.

While our main ambition is to advance the algorithmic foundations of molecular simulation, a major challenge will be to ensure that the theoretical and algorithmic developments will change the fate of applications, as illustrated by our case studies. To foster such a symbiotic relationship between theory, algorithms and simulation, we will pursue high quality software development and integration within the SBL, and will also take the appropriate measures for the software to be widely adopted.

3.4 Applications: modeling interfaces, contacts, and interactions

Keywords: Protein interactions, protein complexes, structure/thermodynamics/kinetics prediction.

Our methods will be validated on various systems for which flexibility operates at various scales. Examples of such systems are antibody-antigen complexes, (viral) polymerases, (membrane) transporters.

Even very complex biomolecular systems are deterministic in prescribed conditions (temperature, pH, etc), demonstrating that despite their high dimensionality, all d.o.f. are not at play at the same time. This insight suggests three classes of systems of particular interest. The first class consists of systems defined from (essentially) rigid blocks whose relative positions change thanks to conformational changes of linkers; a Newton cradle provides an interesting way to envision such as system. We have recently worked on one such system, a membrane proteins involve in antibiotic resistance (AcrB, see 17). The second class consists of cases where relative positions of subdomains do not significantly change, yet, their intrinsic dynamics are significantly altered. A classical illustration is provided by antibodies, whose binding affinity owes to dynamics localized in six specific loops 14, 15. The third class, consisting of composite cases, will greatly benefit from insights on the first two classes. As an example, we may consider the spikes of the SARS-CoV-2 virus, whose function (performing infection) involves both large amplitude conformational changes and subtle dynamics of the so-called receptor binding domain. We have started to investigate this system, in collaboration with B. Delmas (INRAE).

In ABS, we will investigate systems in these three tiers, in collaboration with expert collaborators, to hopefully open new perspectives in biology and medicine. Along the way, we will also collaborate on selected questions at the interface between CSB and systems biology, as it is now clear that the structural level and the systems level (pathways of interacting molecules) can benefit from one another.

5.1 Footprint of research activities

A tenet of ABS is to carefully analyze the performances of the algorithms designed–either formally or experimentally, so as to avoid massive calculations. Therefore, the footprint of our research activities has remained limited.

5.2 Impact of research results

The scientific agenda of ABS is geared towards a fine understanding of complex phenomena at the atomic/molecular level. While the current focus is rather fundamental, as explained in Research program, an overarching goal for the current period (i.e. 12 years) is to make significant contributions to important problems in biology and medicine.

7.1 New software

7.2 Open data

The ongoing collaboration with the Computational Structural Biology team of NINDS (NIH) in Bethesda, MD (USA) is continuing developing the EncoMPASS database for relating membrane protein structures and symmetries. EncoMPASS is the object of a novel recent publication 18 and is undergoing a significant expansion in view of the developing AI tools for structural molecular biology. We propose EncoMPASS as a very reliable source of information on membrane proteins, especially suitable for benchmarking and training prediction algorithms.

8.1 Modeling the dynamics of proteins

Keywords: Protein flexibility, protein conformations, collective coordinates, conformational sampling, loop closure, kinematics, dimensionality reduction.

8.2 Algorithmic foundations

Keywords: Computational geometry, computational topology, optimization, graph theory, data analysis, statistical physics.

8.3 Applications in structural bioinformatics and beyond

Keywords: Docking, scoring, interfaces, protein complexes, phylogeny, evolution.

9.1 International research visitors

9.2 National initiatives

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

11.1 Major publications

• 1 articleJ.-C.Jean-Claude Bermond, D.Dorian Mazauric, V.Vishal Misra and P.Philippe Nain. Distributed Link Scheduling in Wireless Networks.Discrete Mathematics, Algorithms and Applications1252020, 1-38HALDOI
• 2 articleJ.-D.Jean-Daniel Boissonnat and D.Dorian Mazauric. On the complexity of the representation of simplicial complexes by trees.Theoretical Computer Science617February 2016, 17HALDOIback to text
• 3 articleJ.J. Carr, D.D. Mazauric, F.F. Cazals and D. J.D. J. Wales. Energy landscapes and persistent minima.The Journal of Chemical Physics14452016, 4URL: https://www.repository.cam.ac.uk/handle/1810/253412DOIback to text
• 4 articleF.F. Cazals, T.T. Dreyfus, D.D. Mazauric, A.A. Roth and C.C.H. Robert. Conformational Ensembles and Sampled Energy Landscapes: Analysis and Comparison.J. of Computational Chemistry36162015, 1213--1231URL: https://hal.archives-ouvertes.fr/hal-01076317DOIback to text
• 5 articleF.Frédéric Cazals and T.Tom Dreyfus. The Structural Bioinformatics Library: modeling in biomolecular science and beyond.Bioinformatics338April 2017HALDOIback to text
• 6 reportF.Frédéric Cazals and T.Tom Dreyfus. The Structural Bioinformatics Library: modeling in biomolecular science and beyond.RR-8957InriaOctober 2016HAL
• 7 inproceedingsF.Frédéric Cazals and A.Alix Lhéritier. Beyond Two-sample-tests: Localizing Data Discrepancies in High-dimensional Spaces.IEEE/ACM International Conference on Data Science and Advanced AnalyticsIEEE/ACM International Conference on Data Science and Advanced AnalyticsIEEE/ACM International Conference on Data Science and Advanced AnalyticsParis, FranceMarch 2015, 29HALback to text
• 8 inproceedingsF.Frédéric Cazals and A.Alix Lhéritier. Low-Complexity Nonparametric Bayesian Online Prediction with Universal Guarantees.NeurIPS 2019 - Thirty-third Conference on Neural Information Processing SystemsVancouver, CanadaDecember 2019HALback to text
• 9 articleF.Frédéric Cazals, D.Dorian Mazauric, R.Romain Tetley and R.Rémi Watrigant. Comparing Two Clusterings Using Matchings between Clusters of Clusters.ACM Journal of Experimental Algorithmics241December 2019, 1-41HALDOIback to text
• 10 inproceedingsA.Augustin Chevallier, F.Frédéric Cazals and P.Paul Fearnhead. Efficient computation of the volume of a polytope in high-dimensions using Piecewise Deterministic Markov Processes.AISTATS 2022 - 25th International Conference on Artificial Intelligence and StatisticsVirtual, FranceMarch 2022HAL
• 11 articleA.Augustin Chevallier and F.Frédéric Cazals. Wang-Landau Algorithm: an adapted random walk to boost convergence.Journal of Computational Physics4102020, 109366HALDOI
• 12 articleN.Nathann Cohen, F.Frédéric Havet, D.Dorian Mazauric, I.Ignasi Sau Valls and R.Rémi Watrigant. Complexity dichotomies for the Minimum F -Overlay problem.Journal of Discrete Algorithms52-53September 2018, 133-142HALDOIback to text
• 13 articleA.Alix Lhéritier and F.Frédéric Cazals. A Sequential Non-Parametric Multivariate Two-Sample Test.IEEE Transactions on Information Theory645May 2018, 3361-3370HALback to text
• 14 reportS.Simon Marillet, P.Pierre Boudinot and F.Frédéric Cazals. High Resolution Crystal Structures Leverage Protein Binding Affinity Predictions.RR-8733InriaMarch 2015HALback to text
• 15 articleS.Simon Marillet, M.-P.Marie-Paule Lefranc, P.Pierre Boudinot and F.Frédéric Cazals. Novel Structural Parameters of Ig–Ag Complexes Yield a Quantitative Description of Interaction Specificity and Binding Affinity.Frontiers in Immunology8February 2017, 34HALDOIback to text
• 16 articleA.A. Roth, T.T. Dreyfus, C.C.H. Robert and F.F. Cazals. Hybridizing rapidly growing random trees and basin hopping yields an improved exploration of energy landscapes.J. Comp. Chem.3782016, 739--752URL: https://hal.inria.fr/hal-01191028DOIback to text
• 17 articleM.Méliné Simsir, I.Isabelle Broutin, I.Isabelle Mus‐Veteau and F.Frédéric Cazals. Studying dynamics without explicit dynamics: A structure‐based study of the export mechanism by AcrB.Proteins - Structure, Function and BioinformaticsSeptember 2020HALDOIback to text

11.2 Publications of the year

11.3 Cited publications

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• 26 articleK.K.D. Ball and R.R.S. Berry. Dynamics on statistical samples of potential energy surfaces.The Journal of chemical physics11151999, 2060--2070back to text
• 27 bookH.H.B. Callen. Thermodynamics and an Introduction to Thermostatistics.Wiley1985back to text
• 28 articleL.L. Cao, I.I. Goreshnik, B.B. Coventry, J.J.B. Case, L.L. Miller, L.L. Kozodoy, R.R. Chen, L.L. Carter, A.A. Walls, Y.-J.Y-J. Park, E.-M.E-M Strauch, L.L. Stewart, M.M.S. Diamond, D.D. Veesler and D.D. Baker. De novo design of picomolar SARS-CoV-2 miniprotein inhibitors.Science37065152020, 426--431back to textback to textback to textback to text
• 29 articleJ.J. Carr, D.D. Mazauric, F.F. Cazals and D. J.D. J. Wales. Energy landscapes and persistent minima.The Journal of Chemical Physics14452016, 4URL: https://www.repository.cam.ac.uk/handle/1810/253412DOIback to text
• 30 articleB.B. Cousins and S.S. Vempala. A practical volume algorithm.Mathematical Programming Computation822016, 133--160back to text
• 31 bookD.D. Frenkel and B.B. Smit. Understanding molecular simulation.Academic Press2002back to text
• 32 articleR.R. Kannan, L.L. Lovász and M.M. Simonovits. Random walks and an $O^{*}(n^5$) volume algorithm for convex bodies.Random Structures & Algorithms1111997, 1--50back to text
• 33 bookD.D. Landau and K.K. Binder. A guide to Monte Carlo simulations in statistical physics.Cambridge university press2014back to text
• 34 bookT.T. Lelièvre, G.G. Stoltz and M.M. Rousset. Free energy computations: A mathematical perspective.World Scientific2010back to text
• 35 articleC.C. Schön and M.M. Jansen. Prediction, determination and validation of phase diagrams via the global study of energy landscapes.Int. J. of Materials Research10022009, 135back to textback to text
• 36 articleA.A. Senior, R.R. Evans, J.J. Jumper, J.J. Kirkpatrick, L.L. Sifre, T.T. Green, C.C. Qin, A.A. Żídek, A.A. Nelson, A.A. Bridgland, H.H. Penedones, S.S. Petersen, K.K. Simonyan, S.S. Crossan, K.K. Pushmeet, D.D. Jones, D.D. Silver, K.K. Kavukcuoglu and D.D. Hassabis. Improved protein structure prediction using potentials from deep learning.Nature2020, 1--5back to text
• 37 articleD. E.D. E. Shaw, P.P. Maragakis, K.K. Lindorff-Larsen, S.S. Piana, R. O.R. O. Dror, M. P.M. P. Eastwood, J. A.J. A. Bank, J. M.J. M. Jumper, J. K.J. K. Salmon, Y.Y. Shan and W.W. Wriggers. Atomic-level characterization of the structural dynamics of proteins..Science33060022010, 341--346URL: http://dx.doi.org/10.1126/science.1187409back to textback to text
• 38 bookD. J.D. J. Wales. Energy Landscapes.Cambridge University Press2003back to textback to text
• 39 articleL.-P.Lee-Ping Wang, T. J.Todd J Martinez and V. S.Vijay S Pande. Building force fields: an automatic, systematic, and reproducible approach.The journal of physical chemistry letters5112014, 1885--1891back to text