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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 On the one hand, genome sequencing projects give supply protein sequences, and ~200 millions of sequences have been archived in UniProtKB/TrEMBL – which collects the protein sequences yielded by genome sequencing projects.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 ∼108 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 ≫1012 conformations/frames, a (brute) tour de force rarely achieved  34.

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 33, 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.

Figure1: The synergy modeling – experiments, and challenges faced in CSB: illustration on the problem of designing miniproteins blocking the entry of SARS-CoV-2 into cells. From 25. Of note: the first step of the infection by SARS-CoV-2 is the attachment of its receptor binding domain of its spike (RBD, blue molecule), to a target protein found on the membrane of our cells, ACE2 (orange molecule). A strategy to block infection is therefore to engineer a molecule binding the RBD, preventing its attachment to ACE2. (A) Design of a helical protein (orange) mimicking a region of the ACE2 protein. (B) Assessment of binding modes (conformation, binding energies) of candidate miniproteins neutralizing the RBD.

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 22. 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), 25). 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 25, 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  25. 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.

Figure2: The main challenges of molecular simulation: Finding significant local minima of the energy landscape, computing statistical weights of catchment basins by integrating Boltzmann’s factor, and identifying transitions. Practically, d>100 .

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 ℝd to ℝ providing a potential energy for each conformation  35, 32. 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 ∼1,000 ), fitted to reproduce physico-chemical properties of (bio-)molecules  36.

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 34, 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 28. 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  23; fortunately, they enjoy a multi-scale structure  26. 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 ∼ 15 and ∼ 4 orders of magnitude in time and amplitude respectively 21. 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 Kon and Koff ), or fluorescence based methods (for kinetics of folding).

Last activity report : 2023

Results

New results

Modeling the dynamics of proteins

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

Enhanced conformational exploration of protein loops using a global parameterization of the backbone geometry

Flexible loops are paramount to protein functions, with action modes ranging from localized dynamics contributing to the free energy of the system, to large amplitude conformational changes accounting for the repositioning whole secondary structure elements or protein domains. However, generating diverse and low energy loops remains a difficult problem.

This work 18 introduces a novel paradigm to sample loop conformations, in the spirit of the Hit-and-Run (HAR) Markov chain Monte Carlo technique. The algorithm uses a decomposition of the loop into tripeptides, and a novel characterization of necessary conditions for Tripeptide Loop Closure to admit solutions. Denoting m the number of tripeptides, the algorithm works in an angular space of dimension 12m . In this space, the hyper-surfaces associated with the aforementioned necessary conditions are used to run a HAR-like sampling technique. On classical loop cases up to 15 amino acids, our parameter free method compares favorably to previous work, generating more diverse conformational ensembles. We also report experiments on a 30 amino acids long loop, a size not processed in any previous work.

Algorithmic foundations

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

Geometric constraints within tripeptides and the existence of tripeptide reconstructions

Designing movesets providing high quality protein conformations remains a hard problem, especially when it comes to deform a long protein backbone segment, and a key building block to do so is the so-called tripeptide loop closure (TLC) 17. Consider a tripeptide whose first and last bonds ( N1Cα;1 and Cα;3C3 ) are fixed, and so are all internal coordinates except the six {(ϕ,ψ)}i=1,2,3 dihedral angles associated to the three Cα carbons. Under these conditions, the TLC algorithm provides all possible values for these six dihedral angles–there exists at most 16 solutions. TLC moves atoms up to ∼5Å in one step and retains low energy conformations, whence its pivotal role to design move sets sampling protein loop conformations.

In this work 17, we relax the previous constraints, allowing the last bond ( Cα;3C3 ) to freely move in 3D space–or equivalently in a 5D configuration space. We exhibit necessary geometric constraints in this 5D space for TLC to admit solutions. Our analysis provides key insights on the geometry of solutions for TLC. Most importantly, when using TLC to sample loop conformations based on m consecutive tripeptides along a protein backbone, we obtain an exponential gain in the volume of the 5m -dimensional configuration space to be explored.

Applications in structural bioinformatics and beyond

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

Discriminating physiological from non-physiological interfaces in structures of protein complexes: A community-wide study

Community contribution in the scope of the Elixir / 3D Bioinfo project, see the paper and benchmark.

Reliably scoring and ranking candidate models of protein complexes and assigning their oligomeric state from the structure of the crystal lattice represent outstanding challenges. A community-wide effort was launched to tackle these challenges 19. The latest resources on protein complexes and interfaces were exploited to derive a benchmark dataset consisting of 1677 homodimer protein crystal structures, including a balanced mix of physiological and non-physiological complexes. The non-physiological complexes in the benchmark were selected to bury a similar or larger interface area than their physiological counterparts, making it more difficult for scoring functions to differentiate between them. Next, 252 functions for scoring protein-protein interfaces previously developed by 13 groups were collected and evaluated for their ability to discriminate between physiological and non-physiological complexes. A simple consensus score generated using the best performing score of each of the 13 groups, and a cross-validated Random Forest (RF) classifier were created. Both approaches showed excellent performance, with an area under the Receiver Operating Characteristic (ROC) curve of 0.93 and 0.94, respectively, outperforming individual scores developed by different groups. Additionally, AlphaFold2 engines recalled the physiological dimers with significantly higher accuracy than the non-physiological set, lending support to the reliability of our benchmark dataset annotations. Optimizing the combined power of interface scoring functions and evaluating it on challenging benchmark datasets appears to be a promising strategy.

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