Characterization of cell division control strategies through continuous rate models
Cesar Augusto Nieto-Acuna, Juan Carlos Arias-Castro, Carlos Arturo Sanchez-Isaza, Cesar Augusto Vargas-Garcia, Juan Manuel Pedraza
aa r X i v : . [ q - b i o . CB ] M a y Characterization of cell division control strategies through continuous rate models
Cesar Nieto-Acuna, ∗ Juan Arias-Castro,
Carlos Sanchez-Isaza,
Cesar Vargas-Garcia, and Juan Manuel Pedraza † Department of Physics, Universidad de los Andes, Bogot´a, Colombia. South America Department of Systems Biology, Harvard Medical School, Boston, Massachusetts 02115, USA. Department of Mathematics and Engineering, Fundaci´on universitaria Konrad Lorenz, Bogota, Colombia, South America (Dated: June 3, 2019)Recent experiments have supported the
Adder model for
E. coli division control. This model positsthat bacteria grow, on average, a fixed size before division. It also predicts decorrelation betweenthe noise in the added size and the size at birth. Here we use new experiments and a theoreticalapproach based on continuous rate models to explore deviations from the adder strategy, specifically,the division control of
E. coli growing with glycerol as carbon source. In this medium, the divisionstrategy is sizer-like , which means that the added size decreases with the size at birth. We foundthat in a sizer-like strategy the mean added size decreases with the size at birth while the noisein added size increases. We discuss possible molecular mechanisms underlying this strategy, andpropose a general model that encompasses the different division strategies.
Bacterial homeostasis, the control of cell size distri-bution over a population of cells, has been extensivelystudied[1, 2]. Determining the underlying mechanisms isimportant not only for a fundamental understanding ofcell growth, but also because most forms of signaling in-side cells depend on concentrations[3], which depend oncell volume, which in turn will fluctuate in a manner thatdepends strongly on the timing of cell division and itsvariability[4]. Therefore, having an accurate stochasticmodel of cell division is paramount for predicting pheno-typic variability and controlling intra-cellular circuits.Experimental techniques have enabled high through-put measurements of cell growth dynamics, allowing thestudy of cell division strategies not only in bacteria suchas
Escherichia coli [5],
Bacillus subtilis [6],
Caulobactercrescentus [7] and
Pseudomonas aeruginosa [8] but alsoyeast like
Saccaromyces Cervisiae [9] and
Schizosaccha-romyces pombe [10], and archea [11], among others.Division strategies, this is, how bacteria decide whento split into two descendant bacteria, can be classified onthree main paradigms, each with different possible un-derlying mechanisms: one is the timer strategy, in whicha cell waits a fixed time, on average, and then divides. Asecond paradigm is the adder , in which a cell attemptsto add a fixed size, on average, before dividing [12]. Thethird is the sizer , in which a cell grows until it reaches acertain volume[13].These strategies can be distinguished experimentallythrough measurements of added size vs. size at birth.An adder would have a constant added size by definition,whereas a sizer would produce a slope of -1, given theinversely proportional relationship between birth size andremaining growth needed to reach the desired fixed size.On the other hand timer strategy slope is 1. This hasa fundamental problem: is unable to produce stable cell ∗ [email protected] † [email protected] size distributions when cells grow exponentially [14] andthus is not usually found in this kind of cells. This is nota problem for the adder and sizer models.Biological systems can of course be more complicated,incorporating multiple controls that work in tandem oractivate under different conditions. This leads to the phe-nomenological definition of sizer-like [14] and timer-like mechanisms, based upon the slope of added size vs. sizeat birth. Microorganisms like yeast[13], slow-growing E.coli cells[15] and mycobacteria growing in sub-optimalgrowth media[16] have been suggested as examples of sizer-like behavior.Despite recent proposals [17–19], we still lack a mech-anistic understanding of the biochemical mechanisms be-hind division control and how they depend on environ-mental conditions. One approach is to find the genes in-volved through traditional mutation assays[20], but an-other is to obtain a mechanistic model whose behaviormatches experiments, and use it as a guide for whichkinds of molecules to look for. Recent attempts at amechanistic explanation include a threshold on the num-ber of some precursor of division in the cell[21, 22].The main idea behind this proposal is the modelingthe cell decision through continous rate models (CRMs).These models consider not just discrete division events,but the continuous cell cycle. They specify the splittingrate function (SRF)[6], the instantaneous division rate,as a function of physiological parameters such as the cur-rent size, size at birth, growth rate, or the time since lastdivision. Currently, the main problem on CRM is that itis not obvious a priori how to parametrize the SRF[23].To study division control, we use dynamic tracking of
E. coli cells growing in different media in a
Mother Ma-chine microfluidic device[6] (FIG. 1). This device en-ables the imprisonment of cells for measuring their size,growth and gene expression for hundreds of cell lineagesover many generations while allowing continuous mediuminfusion to maintain balanced growth. We observe both adder and sizer-like behavior depending on the media.We propose a biophysical model based on [21], con- (cid:1) m FIG. 1. Left) schematic representation of the mother machine micro-fluid. Right) Actual image in fluorescence channel ofbacteria inside the growth trenches. sisting on a CRM with a SRF which does not dependlinearly on the cell size but on a power of the size. De-pending on the power, all the division paradigms can bemodeled. We then refine our model by comparison withexperiments, looking not only at the added size vs. sizeat birth but also at the noise in the added size. We alsoshow how measuring population-wide dynamics on thou-sands of individual cells allow the use of noise character-istics as additional tools to distinguish between possiblemodels.
I. THEORETICAL CONSIDERATIONS
We assume that each cell-cycle, i.e. the growth be-tween a bacterial division and the next, can be modeledas exponential growth by the system of equations (1):˙ s = µs ˙ τ = 1 , (1)with s the cell size. µ is the growth-rate and τ is thetime elapsed since the previous division, which is resetto 0 after every splitting event. In our experiments, celllength is used as a proxy of the cell size because cell widthis mostly constant[6] and measurements of the area havehigher errors introduced by the small width.Given a cell cycle time τ , the probability of divisionduring the time interval ( τ, τ + dτ ) is described by thecurrent division rate h [24]. Here, we call h the split-ting rate function (SRF) following notation used in otherstudies[5, 6]. By this definition, it can be shown (seeS.M.) that h generates the cumulative distribution func-tion (CDF) P ( τ | s b ) of division at cell cycle time τ giventhe size at birth s b : h ( s ; s b ) = − ddτ ln (1 − P ( τ | s b )) . (2)Thus, if h can be obtained as a function of τ , the in-tegration of (2) can give us the distribution of divisiontimes. With a SRF proportional to a power ( λ ) of the currentcell size ( s ), we have, explicitly: h ( s ; s b ) ≡ ks λ = ks λb exp( λµτ ) . (3)After integration of (3) in (2) (see S.M.), the probabilitydistribution for cell splitting at size s d given its newbornsize s b is: ρ ( s d | s b ) = kµ s λ − d exp (cid:18) − kµλ ( s λd − s λb ) (cid:19) θ ( s d − s b ) (4)with θ ( x ) the Heaviside step function[25]. From (4) every m -th moment can be calculated, resulting in: E [ s md | s b ] = exp (cid:18) kµλ s λb (cid:19) (cid:18) µλk (cid:19) mλ Γ (cid:18) mλ , kµλ s λb (cid:19) (5)where Γ( a, z ) = R ∞ z t a − e − t dt , the incomplete gammafunction.The expected added size E [∆] and the noise in addedsize CV can be obtained using: E [∆ | s b ] = E [ s d | s b ] − s b CV ( s b ) = E [ s d | s b ] − ( E [ s d | s b ]) ( E [ s d | s b ] − s b ) . (6)The dependence of E [∆] on s b is shown in FIG. 2.A.where the three main division strategies can be distin-guished by their corresponding slope. They are: theperfect timer strategy which is obtained when λ → adder when λ → sizer when λ → ∞ . Intermediate strategies are naturally obtainedfor intermediate λ . Timer-like control is obtained when0 < λ < sizer-like control when 1 < λ < ∞ .The typical size ¯ s b , as is shown in FIG. 2.A., is theaverage cell size at birth. Theoretically, this size satisfies: E [ s d | s b = ¯ s b ] = 2 ¯ s b . (7)A closed-form expression for this ¯ s b is not obtained,but it is possible to find it numerically using root-findingalgorithms[25] using (5). In FIG. 2.A., the added sizeand the size at birth are normalized by ¯ s b As it was pointed out in [21] and found in [17], cell divi-sion might require the completion of not one but n events.In our data we see evidence indicating that the noise isfar too low for a single step process (see next section).Therefore, (6) has to be rewritten to take into account n successive but otherwise independent events each witha rate of occurrence nh (see S.M.). The resulting distri-bution is the convolution of n PDFs each following theequation (4). It can be checked that E [ s d | s b ] does notchange appreciably but CV is reduced: CV n = 1 n CV , (8)where CV n is CV for a n-steps mechanism while CV isthe noise for a single step strategy. The relationship CV n vs s b is plotted in FIG 2.D, where each strategy showsa different behavior with s b . While CV n vs s b decreasesfor the timer-like strategy, it increases for the sizer-like strategy and is constant for the adder . Δ / s b λ = 1 $ G G H U λ = . λ = 7 L P H U λ = ∞ 6 L ] H U λ = 7 L P H U O L N H 6 L ] H U O L N H $ 7 K H R U \ λ = 1.0, n = 19 % * O X F R V H $ G G H U λ = 1.5, n = 8 &