Modelling Population Cell Cycle Heterogeneity to Deconvolute Population-Level Measurements

Modelling Population Cell Cycle Heterogeneity to

Deconvolute Population-level Measurements

Melchior du Lac Andrew Scarpelli Joshua Leonard Declan G. Bates

I. INTRODUCTION
Synthetic biology uses mathematical modelling, combined with in silico simulations to guide the design and optimisation of synthetic genetic circuits (SGC). These circuits are usually modelled as a system of ordinary differential equations (ODE). For example, the represillator and the toggle switch, two of the founding SGCâ??s, were successfully designed using this method [1][2]. The models employed in these studies did not, however, predict the level of variations in the oscillatory signals or the desynchronisation observed between even small number of generations when expressed in Escherichia Coli (E.Coli). Even if the molecular mechanisms and biochemistry of the transcription and translation mechanism controlling these SGCâ??s are quite well understood, biological noise dic- tates these non-deterministic expression dynamics [3]. This noise is a combination of the probabilistic nature of the biochemical reactions, called intrinsic noise, and extrinsic noise, that refers to variations in other cellular components that may affect the transcription and translating machinery of the gene or system of interest [4]. For example, in the scope of synthetic biology, the latter noise could arise from cell variability of gene copy numbers from different plasmid copy numbers or from asymmetric cell duplication [5][6][7]. While the effects of intrinsic noise is being increasingly well formalised through stochastic modelling, extrinsic noise is more difficult to elucidate because of the vast number of potential sources and their uncertain influence on the system of interest [3].
To obtain a quantitative understanding of any biological system, intracellular concentrations over time courses are required [8]. However, most -omics technologies sample pop- ulations instead of individual cells. At the population level, the accumulation of biological noise leads isogenic cells to quickly become phenotypically heterogneic, a property that is undesirable for the accurate estimation of expression levels [9]. To reduce the influence of extrinsic noise, basic microbiolog- ical practices dictate that single celled organisms be sampled during exponential growth of culture, to assure that individuals have a similar molecular profile [7]. However, even under these stringent experimental conditions, cells are seldom homoge- neous [10]. This makes the output of such measurements a statistical property of the predominant molecular state of the population, that cannot be used as a reliable estimation of discrete intracellular concentrations [11][8].
Although the origins of heterogeneity are diverse and pre-

Fig. 1. Histogram of the in vitro (red) and in sillico (blue) DNA distributions along the stationary phase of growth in batch culture. The DNA content is normalised by number of chromosomes, and Gaussian spread has been applied to the simulation results using a CV of 0.4.

dominantly still to be elucidated, in some cases, they depend on known biological mechanisms [10][7], such as the cell cycle [12][13]. Indeed, many genes are differentially expressed based on the growth cycle state of the cell [14]. To deconvolve the dynamics of time course data and formalise the influence of cell cycle extrinsic noises on measured sample expression, it is necessary to understand cell cycle heterogeneity in populations of cells. The term deconvolution, in this case, is used not as a strict mathematical term, but to illustrate that population measurements are the sum of individual expression. Using an individual based simulation method to represent the inherent heterogeneity of bacterial populations, combined with a real- istic growth model, we simulate the cell cycle dynamics of a population of cells in conditions of restricted and exponential growth.
II. DRIVING INSTEAD OF DICTATING GROWTH
Because of the multi-dimensional nature of extrinsic noise, it is necessary in the context of modelling frameworks to make assumptions on the state of the cells, and usually takes the form of population â??steady stateâ? [7]. Although this is not equivalent to the systems theory definition, it does imply time invariant properties of the population, where cells synthesise a particular molecule at a constant differential rate [15]. In such
a case, the population is said to be in steady state vis-a-vis the molecule of interest. It is experimentally achieved by growing cells exponentially and in conditions of unrestricted growth (also called balanced growth), where Malthusian growth theory can be applied [16][17]. This established method has been designed to drive populations to behave in a predictable manner in order to reduce the inherent heterogeneity between individual cells [17]. For example, mathematical formalisation of the age distribution in exponentially growing populations, has enabled successful models of population chromosomal distributions for rapidly growing cultures [12]. However the use of Malthusian growth to describe population dynamics and the underlying cellular processes, is not only restricted to exponential experimental conditions, but many examples in the literature show that this is not a robust method to describe slower growth rates [18][19][17]. Worse, the very concept of population steady state is increasingly being challenged [20]. Other growth models such as the Monod equation that prac- tically describes the growth dynamics under nutrient limiting conditions, have no mechanistic biological basis, and so are of limited use in understanding cell growth and the downstream influence on the cellular properties [6].
Since most growth models have limitations in accurately representing the dynamics of population growth, instead of dictating the dynamics of individual cell growth using for example the Monod equation, we drive individual cell growth by population level measurements. Optical density (OD) is a standard method that determines the turbidity of a sample and is used as proxy for the relative changes of mass of a sample. Although the data reported is relative, if one were to determine the true volumetric concentration based on a particular OD, one could normalise OD data that would return time course volumetric changes of a population [8]. Using this estimate of volumetric population concentration, we distribute the volume changes among all members of a population that we are simulating. We term this the â??injection methodâ?, because we are driving the growth of the cell by injecting volume to match the population volumetric concentration. OD data is also used to extract growth growth rates and cell cycle parameters that are of interest to us. However because of we are using the injection method, we cannot infer them from growth models. To perform this without making any assumption of the dynamics of the growth, we use a method qualified as â??instantaneous growth rateâ? [21]. This implies smoothing OD data using the spline interpolation method, enabling the growth dynamics to be accurately extracted from directly measured experimental data [21].
III. MECHANISTIC CELL CYCLE MODEL
Evidence suggests that the dynamics of the cell cycle is tightly correlated with the mass of a cell. Indeed, cells have a constant mass at initiation of DNA replication [22]. This observation formalises why bacterial cells are larger at higher growth rates while being smaller at lower growth rates, and also explains overlapping rounds of replication, where one chromosome may have several origin of replication open if

Fig. 2. Comparison between Keasling et al. Monte Carlo simulation fit to flow cytometry data (left) and injection method growth (right) for a doubling rate of 27 minutes. For both the x-axis represents the DNA content in chromosome numbers and both the y-axis represents the relative cellular concentration. The solid line and the green line both represent the traditional method of deducing DNA population distributions, based on the theoretical age distribution of a population. The noise associated with the green distribution arises from the discrete solution to a continuous function. The solid circles are the simulation results of the Keasling et al. paper and the open circles are experimental data while the blue line is the simulation results of our injection method.

the bacteria is dividing fast enough [22]. We use the Cooper and Helmsetter mechanistic model of the cell cycle that implements the above observation [23], and has been used to model exponentially growing populations [13].
To simulate the inherent heterogeneity of bacterial cell populations, agent based simulations are becoming a more popular method to capture innate social and self emergent behaviours of many aspects of biology [24]. By explicitly representing all members of a populations, simulating them in parallel and implementing some type of communication between agents, the sum of the individuals affect on the system dictates the system behaviour. For our purpose the cells do not communicate, and thus our method can be characterised as individual based.
IV. VALIDATING THE INJECTION METHOD
It has been reported that Monte Carlo simulations of ex- ponentially growing bacterial populations successfully repro- duced the DNA distribution of a bacterial populations [13]. To illustrate that our growth method is valid, we reproduced their simulation conditions using our growth method [13]. We generated a dummy growth curve whoâ??s OD increased with a constant rate and translates to a doubling rate of 27 minutes, as per the authors simulation. The results are presented in Figure 2. It shows that the DNA distribution in the simulation spans from 1.5 to 7 chromosomes, similar to the Keasling et al. simulation and their flow cytometry fit, and that the distribution is skewed to the right. Both are also plotted with the output of the traditional model of the cell cycleâ??s DNA distribution (solid line and green line). It uses the theoretical age distribution, depicted by a probability density function, of a population in exponential growth and returns the relative abundance of the cells with particular DNA concentrations [13].
V. PRELIMINARY RESULTS

To show that our method can replicate the right dynamics of the cell cycle regardless of the growth conditions, we have obtained time course flow cytometry data from the exponential phase to the late stationary phases of growth. To calibrate the data, we arrested the duplication of DNA and fit the corresponding peaks using Gaussian functions and deduced their spread (C V = 0.4). Our initial time course simulation results (Fig. 1) demonstrates the shift from one distribution in the early stationary phase, constituted mostly of cells with two chromosomes, to two distinct ones in the late stationary phase (24h), constituted of mostly one chromosome per cell and a smaller peak at two chromosomes. However the spread and distribution, as seen in vitro, are mostly wrong. We believe that the reason for this are badly conditioned parameters of the cell cycle. In this simulation we used the parameters from the literature that are not only different from the strain of bacteria (TOP10) used in the flow cytometry experiment, but are also extracted during the exponential part of the growth curve [19]. We are currently testing if we can infer more accurate cell cycle parameters from the DNA distribution [19].
Assuming that we can simulate the cell cycle reliably using this method, we plan on extending it to model the effect of plasmid dynamics on SGC functionality. Different plasmid types have different partitioning patterns and understanding the heterogeneic influence of the plasmid copy number and replication would, in much the same way as described for chromosomal genes, lead to differentially expressed genes based on the cell cycle and growth conditions [5][6]. Our method could reliably predict this pattern of heterogeneity and help understand the noise associated with measured expression profiles for plasmids genes as well as chromosomal genes.
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