26 Novembre – Thesis defense - Stefano Polizzi

14 h30 Amphi - Building A9 (Talence campus)

Emergence of log-normal distributions in avalanche processes, validation of 1D stochastic and random network models, with an application to the characterization of cancer cells plasticity.

Many glassy and amorphous materials, like martensites, show characteristic behaviours during constraint induced fractures. These fractures are avalanche processes whose statistics is known to follow in most cases a power-law distribution, reminding of collective behaviour and self-organised criticality. Avalanches of fractures are observed as well in living systems which, if we do not consider active remodelling, can be seen as a glassy network, with a frozen structure.
The actin cytoskeleton (CSK) forms microfilaments organised into higher-order structures by a dynamic assembly-disassembly mechanism with cross-linkers.
Experiments revealed that cells respond to external constraints by a cascade of random and abrupt ruptures of their CSK, suggesting that they behave as a quasi-rigid random network of intertwined filaments. We analyse experimental data on CD34+ cells, isolated from healthy and leukemic bone marrows, however these behaviours have been reproduced on other cells.
Surprisingly, the distribution of the strength, the size and the energy of these rupture events do not follow the power-law statistics typical of critical phenomena and of avalanche size distributions in amorphous materials. In fact, the avalanche size turns out to be log-normal, suggesting that the mechanics of living systems in catastrophic events would not fit
into self-organised critical systems (power-laws).
In order to give an interpretation of this peculiar behaviour we first propose a minimal (1D) stochastic model. This model gives an interpretation of the energy released along the rupture events, in terms of the sum (being energy additive) of a multiplicative cascade process relaxing with time. We distinguish 2 types of rupture events, brittle failures likely corresponding to
irreversible ruptures in a stiff and highly cross-linked CSK and ductile failures resulting from dynamic cross-linker unbindings during plastic deformation without loss of CSK integrity. Our model provides some mathematical and mechanistic understanding of the robustness of the log-normal statistics observed in both brittle and ductile situations. We also show that brittle failures are relatively more prominent in leukemic than in healthy cells, suggesting their greater fragility and their different CSK architecture, stiffer and more reticulated.
This minimal model motivates the more general question of what are the resulting distributions of a sum of correlated random variables coming from a multiplicative process. Therefore, we analyse the distribution of the sum of a generalised branching process evolving with a continuous random reproduction (growth) rate. The process depends only on 2 parameters: the first 2 central moments of the reproduction rate distribution.
We then create a phase diagram showing 3 different regions: 1) a region where the final distribution has all central moments finite and is approximately log-normal. 2) A region where the asymptotic distribution is a power-law, with a decay exponent belonging to the interval [1;3], whose value is uniquely determined by the model parameters. 3) Finally, we found an exact log-normal size, non-stationary, distribution region. In all cases correlations are fundamental.
Increasing the level of complexity for avalanche modelling, we propose then a random Erdös-Rényi network to model a cell CSK, identifying the network nodes as the actin filaments, and its links as actin cross-linkers. On this structure we simulate avalanches of ruptures.
Our simulations show that we can reproduce the log-normal statistics with two simple ingredients: a random network without characteristic length scale, and a breaking rule capturing
the observed visco-elasticity of living cells. This work paves the way for future applications to many phenomena in living systems that include large populations of individual, non-linear, elements (brain, heart, epidemics) where similar log-normal statistics have also been observed.

Event localization