Statistical inference in transport- fragmentation models Marc Statistical inference in transport-fragmentation Hoffmann models Genealogical versus temporal data The size Marc Hoffmann dependent division rate model Paris-Dauphine University Estimating the age dependent Van Dantzig Seminar, 6 March 2015 division rate
Acknowledgements Statistical inference in transport- fragmentation models This talk is based on joint projects (some are still in progress!) Marc Hoffmann with - M. Doumic (INRIA) Genealogical versus temporal - N. Krell (University of Rennes) data - A. Olivier (Paris-Dauphine University) The size dependent division rate - P. Reynaud-Bouret (CNRS) model - V. Rivoirard (Paris-Dauphine University) Estimating the age dependent - L. Robert (INRA) division rate
Context (1/4) Statistical inference in transport- fragmentation models Marc We consider (simple) branching processes with Hoffmann deterministic evolution between jump times. Genealogical Such models appear as toy models for population growth versus temporal in cellular biology. data The size We wish to statistically estimate the parameters of the dependent division rate model, in order to ultimately discriminate between model different hypotheses related to the mechanisms that Estimating the age trigger cell division. dependent division rate
Context (2/4) Statistical inference in transport- fragmentation We structure the model by state variables for each models individual like size, age , growth rate, DNA content and so Marc Hoffmann on. Genealogical The evolution of the particle system is described by a versus common mechanism: temporal data 1 Each particle grows by “ingesting a common nutrient” = The size deterministic evolution . dependent division rate 2 After some time, depending on a structure variable, each model particle gives rise to k = 2 offsprings by cell division = Estimating the age branching event . dependent division rate Our goal in this talk: estimate the branching rate as a function of age or size (or both).
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Evolution of a E. Coli population.
Context (3/4) Statistical inference in transport- fragmentation models Deterministically the density of structured state variables Marc Hoffmann evolves according to a so-called fragmentation-transport PDEs Genealogical versus temporal Stochastically, the particles evolve according to a data piecewise deterministic Markov process that evolves The size dependent along a branching tree. division rate model We study nonparametric inference of the division rate , Estimating with the concern of matching deterministic and stochastic the age dependent approaches. division rate
Context (4/4) Statistical inference in transport- fragmentation models Marc I will follow a “pedestrian route” by reviewing some of the Hoffmann results we progressively obtained by “trial-and-error”. Genealogical In particular, the results are highly sensitive to the choice versus temporal of the observation schemes (genealogical versus temporal). data The size Our control experiments are data sets extracted from the dependent division rate observation of 88 microcolonies of E. Coli bacteria cultures (a model colony is followed from a single ancestor up to a few hundreds Estimating the age descendants). dependent division rate
Outline Statistical inference in transport- fragmentation models Marc 1 Genealogical versus temporal data Hoffmann Genealogical versus temporal 2 The size dependent division rate model data Estimation at a (large) fixed time in a proxy model The size dependent Estimation through genealogical data division rate model Estimating the age 3 Estimating the age dependent division rate dependent division rate
Genealogical representation Statistical inference in transport- In the talk we focus on structuring variables that are either fragmentation models age or size. Marc Hoffmann The population evolution is associated with an infinite marked binary tree Genealogical versus temporal ∞ data � with { 0 , 1 } 0 := ∅ . { 0 , 1 } n U = The size dependent n =0 division rate model To each cell or node u ∈ U , we associate a cell with size Estimating the age at birth given by ξ u and lifetime ζ u . dependent division rate To each u ∈ U , we associate a birth time b u and a time of death d u so that ζ u = d u − b u .
Observation scheme I: temporal data Statistical inference in transport- Fix a (large) T > 0. Define fragmentation models � � U T = u ∈ U , b u ≤ T . Marc Hoffmann We have U T = ˚ Genealogical U T ∪ ∂ U T , with versus temporal � � � � data ˚ U T = u , d u ≤ T and ∂ U T = u , b u ≤ T < d u The size dependent division rate We observe model � � Estimating ζ T and/or ξ T the age u , u ∈ U T u dependent division rate where ζ T u = min { d u , T } − b u , and ξ T u = ξ u if d u ≤ T and the “size of u at time T ” otherwise.
Observation scheme II: genealogical data Statistical | u | = n if u = ( u 1 , . . . , u n ) ∈ U , inference in transport- uv = ( u 1 , . . . , u n , v 1 , . . . , v m ) if v = ( v 1 , . . . , v m ) ∈ U . fragmentation Sparse tree case Given u ( n ) ∈ U , with | u ( n ) | = n , let models Marc � � u ∈ U , uw = u ( n ) for some w ∈ U Hoffmann U u ( n ) = . Genealogical We observe versus temporal � � data ζ u and/or ξ u , u ∈ U u ( n ) . The size dependent division rate model Full tree case For n = 2 k n , define Estimating the age U [ n ] = { u ∈ U , | u | ≤ k n } . dependent division rate We observe � � ξ u and/or ζ u , u ∈ U [ n ] .
Temporal data Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : Genealogical tree observed up to T = 7 for a time-dependent division rate B ( a ) = a 2 ( 60 cells). In blue: ˚ U T . In red: ∂ U T .
Genealogical data Statistical inference in transport- fragmentation models Marc Hoffmann Genealogical versus temporal data The size dependent division rate model Estimating the age dependent division rate Figure : The same outcome organised at a genealogical level.
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