Estimating the division rate from indirect measurements of single cells

Abstract

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? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. 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.

Individual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle

“…Much more realistic is the following case, studied in [111]: Assume we are given x 1 , • • • , x n a n− sample of realizations of X 1 , • • • , X n i.i.d. random variables of density N x k (x) defined by…”

“…S n,h (z) = Proof: As a sketch of the proof in this simpler case where τ (x) = κx (see [111] for a general growth rate τ (x)) we first write the equation for N 1 , noticing that N 1 = x k−1 N k up to a constant, and integrate it along z to find the dilation equation (4.36). We then solve the equation for C(z, x) = κxN 1 (z, x) along the characteristics, and find κxN 1 (z, x) = κ(x − z)N 1 (0, x − z)e − z 0 B(s)ds .…”

Continuous Rate Modelling of bacterial stochastic size dynamics

“…The division rate h(s|s ) in general depends on hidden variables like the number steps but can estimated, at least numerically [29][30][31]. This In the case of single step division, this rate can be written The adder strategy, observed for instance, in E. coli and B. subtilis [3], is considered the most common strategy in bacteria.…”

Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation

“…Statistical evidence of asymmetrical divisions and biological consequences are described in Stewart et al [55]. See also Ackermann et al [1], Aguilaniu et al [2], Banks et al [4], Doumic, Robert and co-authors [54,28] or Moseley [49] for discussions on these topics. The population can be described by a stochastic individual-based (particle) model, where the population at time t is represented by a random measure that is the sum of Dirac masses on R + weighting the cells' sizes.…”

“…Statistical evidence of asymmetrical divisions and biological consequences are described in Stewart et al (2005). See also Ackermann et al (2003), Aguilaniu et al (2003), Banks et al (2011), Robert et al (2014), Doumic et al (2019), or Moseley (2013) for discussions on these topics.…”