# RESEARCH

Synthesis report for the years 2014-2017

Here you can find the general presentation of the team: Mamba presentation – 2014 to 2017

and you may download the synthesis report: MAMBA synthesis report – 2014 to 2017

### Overall Objectives

The MAMBA (Modelling and Analysis in Medical and Biological Applications) team is the continuation of the BANG (Biophysics, Numerical Analysis and Geophysics) team, which itself was a continuation of the former project-team M3N. Historically, the BANG team, headed by Benoît Perthame during 11 years (2003-2013), has developed models, simulations and numerical algorithms for problems involving dynamics of Partial Differential Equations (PDEs).

The dynamics of complex physical or biophysical phenomena involves many agents, e.g. proteins or cells, which can be seen as active agents. Mathematically, they can be represented either explicitly as individuals with their dynamics modelled e.g. through branching trees and piecewise deterministic Markov processes (PDMP), or stochastic differential equations, or under certain conditions be grouped or locally averaged, in which case their dynamics is mimicked by Ordinary or Partial Differential Equations (ODEs/PDEs).

Biology and medicine presently face the difficulty to make sense of the data newly available by means of recent signal acquisition methods and to take appropriate actions through possible treatment pathways. Modeling through agent-based or continuous models is a unique way to explain (model) the observations and then compute, control and predict the consequences of the mechanisms under study. These are the overall goals of Mamba.

## Results

#### Modelling Polymerization Processes

In 2017, we evidenced the presence of several polymeric species by using data assimilation methods to fit experimental data from H. Rezaei’s lab  [article]; new experimental evidence reinforced these findings  [article], [article]. The challenges are now to propose mathematical models capable of tracking such diversity while keeping sufficient simplicity to be tractable to analysis.

In collaboration with Klemens Fellner from the university of Graz, we propose a new model, variant of the Becker-Döring system but containing two monomeric species, capable of displaying sustained though damped oscillations as is experimentally observed. We also proposed a statistical test to validate or invalidate the presence of oscillations in experimental highly nonstationary signals  [article].

#### Asymptotic behaviour of structured-population equations

Pierre Gabriel and Hugo Martin studied the mathematical properties of a model of cell division structured by two variables – the size and the size increment – in the case of a linear growth rate and a self-similar fragmentation kernel  [article]. They first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue 1 from a solution of a certain one dimensional fixed point problem. Then they prove the existence and uniqueness of this fixed point in the appropriate ${L}^{1}$ weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.

Etienne Bernard, Marie Doumic and Pierre Gabriel proved in  [article] that for the growth-fragmentation equation with fission into two equal parts and linear growth rate, under fairly general assumptions on the division rate, the solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypo-coercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted ${L}^{2}$ space, where well-posedness is obtained via semigroup analysis. They also propose a non-dissipative numerical scheme, able to capture the oscillations.

Pierre Gabriel and Hugo Martin then extended this asymptotic result in the framework of measure solutions  [article]. To do so they adopt a duality approach, which is also well suited for proving the well-posedness when the division rate is unbounded. The main difficulty for characterizing the asymptotic behavior is to define the projection onto the subspace of periodic (rescaled) solutions. They achieve this by using the generalized relative entropy structure of the dual problem.

#### Marie Doumic and Adélaïde Olivier

Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? The article  [article] adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model – the model studied by Hugo Martin and Pierre Gabriel in  [article]. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

#### Ph. Robert and G. Vignoud

Synaptic plasticity is a common mechanism used to model learning in stochastic neural networks, STDP is a great example of such mechanisms. We develop a simple framework composed by two neurons and one synaptic weight, seen as stochastic processes and study the existence and stability of such distributions, for a wide range of classical synaptic plasticity models. Using two simple examples of STDP, the calcium-based rule and the all-to-all pair-based rule, we apply stochastic averaging principles and obtain differential equations for the limit processes, based on the invariant distributions of the fast system when the slow variables are considered fixed. We study a general stochastic queue to approximate the calcium-based rule and are able to have an analytical solution for the invariant distribution of the fast synaptic processes. We also detail some simpler systems, either through some approximations or simulations to put into light the influences of different biologically-linked parameters on the dynamics of the synaptic weight.

#### G. Vignoud. Collaboration with J. Touboul (Brandeis University)

Spike-Timing Dependent Plasticity (STDP) in the striatum is viewed as a substrate for procedural learning. Striatal projecting neurons (SPNs) express anti-Hebbian plasticity at corticostriatal synapses, (a presynaptic cortical spike followed by a postsynaptic striatal spike leads to the weakening of the connection, whereas the reverse pairing leads to potentiation ). SPNs need to integrate many inputs to spike, and as such, their main role is to integrate context elements to choose between different sensorimotor associations. In this work, we develop a simple numerical model of the striatum, integrating cortical spiking inputs to study the role of anti-Hebbian STDP in pattern recognition and sequence learning. Cortical neurons are seen as binary input neurons and one striatal SPN is modeled as a leaky-integrate-and-fire neuron. Combined informations from the output, reward and timing between the different spikes modify the intensity of each connection, through two mechanisms: anti-Hebbian STDP and dopaminergic signaling, using three-factor learning rules. We have added a second output neuron with collateral inhibition which leads to an improvement of the global accuracy. In another project, we studied the dynamics of learning, by shutting off/on the dopaminergic plasticity, and compare it to DMS/DLS experimental and behavioral experiments. We show that anti-Hebbian STDP favors the learning of complete sequence of spikes, such as is needed in the striatum, whereas, even if Hebbian STDP helps to correlate the spiking of two connected neurons, it is not sufficient to integrate of long sequences of correlated inputs spikes.

#### G. Vignoud. Collaboration, with Team Venance (CIRB/Collège de France)

Striatal medium spiny neurons (MSNs) are segregated into two subpopulations, the D1 receptor-expressing MSNs (the direct striatonigral pathway) and the D2 receptorexpressing MSNs (the indirect striatopallidal pathway). The fundamental role of MSNs as output neurons of the striatum, and the necessary distinction between D1- and D2-expressing neurons accentuate the need to clearly distinguish both subpopulations in electrophysiological recordings in vitro and in vivo. Currently, fluorescent labelling of the dopaminergic receptors in mice enables a clear differentiation. However, multiplying in vivo the number of genetic markers (optogenetics, fluorescence) hinders possibilities for other genetic manipulations. Moreover, electrophysiological properties of fluorescents neurons can slightly differ from “native” cells and false-positive can be observed. The lack of a proper way to separate D1- and D2-MSNs based on electrophysiological properties led us to devise a detection algorithm based on action potential profile. We used more than 450 D1/D2 labelled MSNs from in vitro patch-clamp recordings (different experimentalists, different setups and protocols), to characterize and identify properties that facilitate the MSN discrimination. After analyzing passive and active MSN membrane properties, we built an extensive dataset and fed it into classical machine learning classification methods. The training of the different algorithms (k-nearest neighbors, random forest, deep neural networks, …) was performed with the scikit-learn Python library, and the optimized classifier was able to correctly discriminate neurons in the dorsolateral striatum at 76% (and up to 83% if we allow the classifier to reject some MSNs). This study developed an efficient classification algorithm for D1/D2-MSNs, facilitating cell discrimination without specific genetic fluorescent labelling, leaving some room for other genetic markers and optogenetic labeling.

#### Ph. Robert and G. Vignoud

We have investigated the asymptotic properties of self-interacting point processes introduced by Kerstan (1964) and Hawkes and Oakes (1974). These point processes have the property that the intensity at some point $t\in \left(–\infty ,+\infty \right)$ is a functional of all points of the point process before $t$ . Such a process is said to be stable if it has a version whose distribution is invariant by translation. By using techniques of coupling and Markovian methods, we have been able to obtain some existence and uniqueness results with weaker conditions than in the current literature.

#### Ph. Robert

In Robert  [article] we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, the production process of proteins. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.

#### Marie Doumic, Dietmar Oelz, Alex Mogilner

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of the article  [article] is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.

#### Control Strategies for Sterile Insect Techniques

We proposed different models to serve as a basis for the design of control strategies relying on releases of sterile male mosquitoes (Aedes spp) and aiming at elimination of wild vector population. Different types of releases were considered (constant, periodic or impulsive) and sufficient conditions to reach elimination were provided in each case [article]We also estimated sufficient and minimal treatment times. A feedback approach was introduced, in which the impulse amplitude is chosen as a function of the actual wild population [article].

#### Optimal replacement strategies, application to Wolbachia

We modelled and designed optimal release control strategy with the help of a least square problem. In a nutshell, one wants to minimize the number of uninfected mosquitoes at a given time horizon, under relevant biological constraints. We derived properties of optimal controls and studied a limit problem providing useful asymptotic properties of optimal controls [article], [article].

#### Oscillatory regimes in population models

Understanding mosquitoes life cycle is of great interest presently because of the increasing impact of vector borne diseases. Observations yields evidence of oscillations in these populations independent of seasonality, still unexplained. We proposed [article] a simple mathematical model of egg hatching enhancement by larvae which produces such oscillations that conveys a possible explanation.

On the other hand, population oscillations may be induced by seasonal changes. We considered a biological population whose environment varies periodically in time, exhibiting two very different “seasons”, favorable and unfavorable. We addressed the following question: the system’s period being fixed, under what conditions does there exist a critical duration above which the population cannot sustain and extincts, and below which the system converges to a unique periodic and positive solution? We obtained [article], [article] sufficient conditions for such a property to occur for monotone differential models with concave nonlinearities, and applied the obtained criterion to a two-dimensional model featuring juvenile and adult insect populations.

#### Feedback control principles for population replacement by Wolbachia

The issue of effective scheduling of the releases of Wolbachia-infected mosquitoes is an interesting problem for Control theory. Having in mind the important uncertainties present in the dynamics of the two populations in interaction, we attempted to identify general ideas for building release strategies, which should apply to several models and situations [article]. These principles were exemplified by two interval observer-based feedback control laws whose stabilizing properties were demonstrated when applied to a model retrieved from [article].

#### Bacterial motion by Rerun and tumble

Collective motion of chemotactic bacteria such as Escherichia coli relies, at the individual level, on a continuous reorientation by runs and tumbles. It has been established that the length of run is decided by a stiff response to a temporal sensing of chemical cues along the pathway. We describe a novel mechanism for pattern formation stemming from the stiffness of chemotactic response relying on a kinetic chemotaxis model which includes a recently discovered formalism for the bacterial chemotaxis [article]. We prove instability both for a microscopic description in the space-velocity space and for the macroscopic equation, a flux-limited Keller-Segel equation, which has attracted much attention recently. A remarkable property is that the unstable frequencies remain bounded, as it is the case in Turing instability. Numerical illustrations based on a powerful Monte Carlo method show that the stationary homogeneous state of population density is destabilized and periodic patterns are generated in realistic ranges of parameters. These theoretical developments are in accordance with several biological observations.

This motivates also our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term [article]. Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an un- stable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response. In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counter-balancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.

Another activity concerns the relation between the tumbling rate and the internal state of bacteria. The study [article] aims at deriving models at the macroscopic scale from assumptions on the microscopic scales. In particular we are interested in comparisons between the stiffness of the response and the adaptation time. Depending on the asymptotics chosen both the standard Keller-Segel equation and the flux-limited Keller-Segel (FLKS) equation can appear. An interesting mathematical issue arises with a new type of equilibrium equation leading to solution with singularities.

#### Numerical methods for cell aggregation by chemotaxis

Three-dimensional cultures of cells are gaining popularity as an in vitro improvement over 2D Petri dishes. In many such experiments, cells have been found to organize in aggregates. We present new results of three- dimensional in vitro cultures of breast cancer cells exhibiting patterns. Understanding their formation is of particular interest in the context of cancer since metastases have been shown to be created by cells moving in clusters. In the paper [article], we propose that the main mechanism which leads to the emergence of patterns is chemotaxis, i.e., oriented movement of cells towards high concentration zones of a signal emitted by the cells themselves. Studying a Keller-Segel PDE system to model chemotactical auto-organization of cells, we prove that it is subject to Turing instability if a time-dependent condition holds. This result is illustrated by two-dimensional simulations of the model showing spheroidal patterns. They are qualitatively compared to the biological results and their variability is discussed both theoretically and numerically.

This motivates to study parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes that are shown to preserve, at the discrete level, the fundamental properties of the solutions, namely energy dissipation, steady states, positivity and conservation of total mass [article]. These requirements happen to be critical when it comes to distinguishing between discrete steady states, Turing unstable transient states, numerical artifacts or approximate steady states as obtained by a simple upwind approach. These schemes are obtained either by following closely the gradient flow structure or by a proper exponential rewriting inspired by the Scharfetter-Gummel discretization. An interesting fact is that upwind is also necessary for all the expected properties to be preserved at the semi-discrete level. These schemes are extended to the fully discrete level and this leads us to tune precisely the terms according to explicit or implicit discretizations. Using some appropriate monotonicity properties (reminiscent of the maximum principle), we prove well-posedness for the scheme as well as all the other requirements. Numerical implementations and simulations illustrate the respective advantages of the three methods we compare.

#### Focus on cancer

Modelling Acute Myeloid Leukaemia (AML) and its control by anticancer drugs by PDEs and Delay Differential equations

This theme has continued to be developed in collaboration with Catherine Bonnet, Inria DISCO (Saclay)  [article], [article]. Without control by drugs, but with representation of mutualistic interactions between tumor cells and their surrounding support stroll cells, it has also, in collaboration with Delphine Salort and Thierry Jaffredo (LCQB-IBPS) given rise to a recent work by Thanh Nam Nguyen, hired as HTE and ERC postdoctoral fellow at LCQB, submitted as full article  [article].

Adaptive dynamics setting to model and circumvent evolution towards drug resistance in cancer by optimal control

The research topic “Evolution and cancer”, designed in the framework of adaptive dynamics to represent and overcome acquired drug resistance in cancer, initiated in  [article], [article] and later continued in  [article], [article], [article], has been recently summarised in  [article] and has been the object of the PhD thesis work of Camille Pouchol, see above “Cell population dynamics and its control” . It is now oriented, thanks to work underway by Cécile Carrère, Jean Clairambault, Tommaso Lorenzi and Grégoire Nadin, in particular towards the mathematical representation of bet hedging in cancer, namely a supposed optimal strategy consisting for cancer cell populations under life-threatening cell stress in diversifying their phenotypes according to several resistance mechanisms, such as overexpression of ABC transporters (P-glycoprotein and many others), of DNA repair enzymes or of intracellular detoxication processes. According to different deadly insults the cancer cell population is exposed to, some phenotypes may be selected, any such successful subpopulation being able to store the cell population genome (or subclones of it if the cell population is already genetically heterogeneous) and make it amenable to survival and renewed replication.

Philosophy of cancer biology

This new research topic in Mamba, dedicated to explore possibly underinvestigated, from the mathematical modelling point of view, parts of the field of cancer growth, evolution and therapy, has been the object of a presentation by Jean Clairambault at the recent workshop “Philosophy of cancer biology’

https://www.philinbiomed.org/event/philosophy-of-cancer-biology-workshop/.

This workshop gathered most members worldwide of this small, but very active in publishing, community of philosophers of science whose field of research is “philosophy of cancer”, as they call it themselves. This topic offers a clear point of convergence between mathematics, biology and social and human sciences.

#### Deformable Cell Modeling: biomechanics and Liver regeneration

• Biomechanically mediated growth control of cancer cells The key intriguing novelty was that the same agent-based model after a single parameter has been calibrated with growth data for multicellular spheroids without application of external mechanical stress by adapting a single parameter, permitted to correctly predict the growth speed of multicellular spheroids of 5 different cell lines subject of external mechanical stress. Hereby the same mechanical growth control stress function was used without any modification [article]. The prediction turned out to be correct independent of the experimental method used to exert the stress, whereby once a mechanical capsule has been used, once dextran has been used in the experiments.

• Regeneration of liver with the Deformable Cell Model. The key novelty was the implementation of the model itself, but an interesting novel result is that the DCM permits closure of a pericentral liver lobule lesion generated by drug-induced damage with about 5 times smaller active migration force due to the ability of the cell to strongly deform and squeeze into narrow spaces between the capillaries. This finding stresses that a precise mechanical description is important in view of quantitatively correct modeling results [article]. The deformable cell model however could be used to calibrate the interaction forces of the computationally much cheaper center-based model to arrive at almost the same results.