INferring Anomalous Diffusions in LIving Cells from single particle tracking experiments – INADILIC

The project combines three complementary approaches:
1) theoretical tools such as probabilistic and statistical methods, analysis of partial differential equations, asymptotic and spectral methods, and properties of special functions of mathematical physics;
2) numerical simulations including Monte Carlo simulations, finite elements method, generation of various random numbers and Gaussian processes;
3) analysis of experimental data acquired by external collaborators of the project.
Theoretical tools allow one to characterize the distributions of inferred quantities in order to construct the best statistical estimators adapted to different types of anomalous diffusion. Numerical simulations serve to analyze and to validate these estimators on trajectories that are artificially generated and thus fully controlled. Finally, the analysis of experimental trajectories allows one to identify the transport mechanisms and the most adequate mathematical models, as well as to yield biophysical interpretations. The combined use of these complementary approaches is crucial for this project.

During the first phase of the project (18 months), the following results have been obtained:
1) In order to perform a robust analysis of experimental trajectories, one needs appropriate models of the intracellular transport. For this purpose, we simulated the motion of a tracer in an overcrowded heterogeneous medium that can model the cytoplasm. We revealed the role of the spatial heterogeneity (ignored so far) and showed the anomalous character of the tracer motion in certain regimes.
2) We started a comparative study of inference techniques. First, the optimal choice of the sliding window in the time average of the quadratic displacement in presence of noise was proposed for different models of anomalous diffusion. Second, the ergodicity test was revised and greatly improved to allow its quantitative application to experimental data.
3) In collaboration with group of M. Tamm (Moscow State University), we studied the crossing (overlap) statistics of two Brownian trajectories and obtained the exact form of the scaling functions.
4) In collaboration with group of O. Bénichou and R. Voituriez (University Paris-6), we resolved analytically the long standing problem of finding the distribution of the exit times from circular domains through holes. In addition, the solution was generalized to biased diffusion in order to account for active transport by motor proteins. This study has been completed by the analysis of the first exit points.
5) Finally, diffusion under a harmonic potential and the associated escape problem were reviewed. This problem is important for a better characterization of the motion of a tracer trapped by a laser beam of optical tweezers. In particular, the escape statistics from a trap can serve to estimate the force exerted by motor proteins and to detect the phases of active transport.

After having obtained the first results mentioned above, we will elaborate the project according to the initially formulated directions, namely, systematic comparison and improvement of existing inference tools, understanding the probabilistic nature of the inferred quantities, accounting for experimental «imperfections«. The advances in these directions should allow to develop a synergetic inference procedure that will be released for a broad scientific community. In a long term perspective, this project will result in further understanding of the physical mechanisms and better mathematical modeling of the cellular transport, with potential bio-medical applications to controlling viral infections and efficient drug delivery in medical treatments.

Publications in peer-reviewed international journals:
S. K. Ghosh, A. G. Cherstvy, D. S. Grebenkov, and R. Metzler, Anomalous, non-Gaussian tracer diffusion in heterogeneously crowded environments (submitted to New J. Phys.); disponible à arxiv.org/abs/1508.02029
J.-F. Rupprecht, O. Benichou, D. S. Grebenkov, R. Voituriez, Exit time distribution in spherically symmetric two-dimensional domains, J. Stat. Phys. 158, 192-230 (2015)
D. S. Grebenkov, First exit times of harmonically trapped particles: a didactic review, J. Phys. A 48, 013001 (2015)
D. S. Grebenkov, Analytical representations of the spread harmonic measure density, Phys. Rev. E 91, 052108 (2015)
A. S. Serov, C. Salafia, M. Filoche, D. S. Grebenkov, Analytical theory of oxygen transfer in the human placenta, J. Theor. Biol. 368, 133-144 (2015)
A. S. Serov, C. Salafia, P. Brownbill, D. S. Grebenkov, M. Filoche, Optimal villous density for maximal oxygen uptake in the human placenta, J. Theor. Biol. 364, 383-96 (2015)
M. V. Tamm, V. I. Stadnichuk, A. M. Ilyina, D. S. Grebenkov Overlap of two Brownian trajectories: Exact results for scaling functions, Phys. Rev. E 89, 042137 (2014)
D. S. Grebenkov, D. V. Nguyen, J.-R. Li, Exploring diffusion across permeable barriers at high gradients. I. Narrow pulse approximation, J. Magn. Reson 248, 153-163 (2014)
D. S. Grebenkov, Exploring diffusion across permeable barriers at high gradients. II. Localization regime, J. Mang. Reson. 248, 164-176 (2014)

Transport of macromolecules, organelles and vesicles in living cells is a very complicated process that essentially determines and controls many biochemical reactions, growth and functioning of cells. The passive thermal diffusion through the overcrowded cytoplasm is combined with the active transport by motor proteins attached to microtubules. This intricate mechanism results in anomalous diffusions that found abundant experimental evidences but no consensus on the physical mechanism and the appropriate mathematical model is achieved so far. Single-particle tracking (SPT) experiments (video-tracking, optical tweezers, etc.) survey individual random trajectories of tracers inside living cells and can thus provide the missing information on the molecular transport in order to discriminate between different physical mechanisms (e.g., “caging” in the overcrowded cellular environment, visco-elastic properties of the cytoskeleton, hierarchical geometrical structure of the cytoplasm and/or nucleus, etc.) and to identify the appropriate theoretical model of anomalous diffusion (e.g., continuous-time random walks, generalized Langevin equation, diffusion on fractals, etc.). In SPT, an ensemble average of the quantities of interest (e.g., diffusivity, viscosity, first passage times, etc.) is often unavailable or even undesired, as tracers move in spatially heterogeneous and time evolving media such as living cells. One faces therefore a challenging problem of inferring dynamical, structural and functional properties of living cells from a limited (small) number of individual random realizations of an unknown stochastic process. The project aims at gathering physical, biological and statistical expertise to build a reliable inference procedure for the analysis of SPT experiments. The project will be developed in several complementary directions:
• A systematic comparison between recently developed inference tools (maximal excursion method, first passage time statistics, local mean-square displacement analysis, Bayesian methods, p-variation tests, etc.) on a common set of simulated/experimental trajectories. Such a systematic study will reveal the limitations/drawbacks of these separate methods, their sensitivity to different diffusion mechanisms and their robustness against variable experimental conditions.
• A theoretical and numerical study of the probabilistic properties of the inferred quantities (such as, e.g., the time-averaged mean-square displacement) for various kinds of anomalous diffusions. The knowledge of their probability distributions will allow for accurate statistical estimations and reliable interpretation of experimental data, with a potential disentanglement of statistical and biological sources of dispersion in measured quantities.
• Accounting for experimental “imperfections” (measurement noises, static localization uncertainty, blurring of a particle's position over camera integration times, presence of a laser trapping potential, etc.) of video-tracking and optical tweezers acquisition, which were mainly ignored so far in theoretical models and inference tools.
Bringing these complementary fields together, we aim at establishing a synergetic inference protocol in which various statistical tools are combined, while their sensitivity and robustness are improved through the knowledge of the probability distributions of the inferred quantities and by accounting for experimental “imperfections”. This protocol will be released as a numerical library to assist experimentalists and biologists in statistical analysis of their data. A reliable biophysical and statistical analysis of SPT experiments will result in further understanding of the physical mechanisms and better mathematical modeling of the cellular transport, with potential bio-medical applications to controlling viral infections and efficient drug delivery in medical treatments.