## Introduction

The cell division cycle coordinates the duplication of all essential cellular components and their separation into two distinct cells. Accurate regulation of cell division is critical and imposed by an intricate regulatory machinery. The eukaryotic cell division cycle is primarily driven by sequential accumulation and destruction of cyclins, which act as activators and targeting subunits for a constitutively present cyclin dependent kinase (CDK) [1, 2]. The cell division cycle's core regulator CDK is itself regulated by external and internal signals, including nutrients, hormones and physical contact with neighbouring cells. This regulation is essential, as inaccurate decisions either to proliferate or to remain quiescent may prove fatal for unicellular and multicellular organisms alike. The mechanistic architecture of the machinery that executes the decision to divide has been studied intensely, giving a good understanding of how duplication and division is orchestrated [3, 4]. In contrast, the actual cue(s) triggering the decision to divide is still unknown. In microorganisms such as yeast where cells have been observed to maintain a certain size, cell size itself, or an equivalent, has been proposed to be the critical cue [5, 6]. However, the mechanistic basis for linking cell cycle events and cell growth is unknown, and hence the nature of the cue(s) driving the progression through the cell division cycle remains a matter of debate [7].

In the premier eukaryotic model organism *Saccharomyces cerevisiae,* cells divide asymmetrically [8]. At the START of the cell division cycle, the mother cell initiates DNA replication and spindle pole duplication and polarizes her actin cytoskeleton to grow a bud by polar secretion through a narrow bud neck. The S phase is followed by a G_{2} phase of comparatively constant duration during which the continued polar growth leads to a volume increase almost exclusively in the bud [8]. Thereafter, the cell enters M phase and initiates rapid mitotic division in which the bud receives a nucleus and separates from the mother. Daughter cells are born at a fraction of the mothers' cell volume and hence need to acquire more mass before they are ready to pass through a subsequent cell division cycle. This is reflected in a prolonged duration of G_{1}, which is recognized as the primary phase for growth regulation of the cell division cycle in *S. cerevisiae* [5, 9]. This differential G_{1} duration is instrumental to maintain the size homeostasis within the population and has been attributed to the regulatory network upstream of START (reviewed in [10]). The earliest undisputed activator of this G_{1} network is the cyclin Cln3, which binds to and activates the CDK Cdc28. Cdc28-Cln3 phosphorylates Whi5 to relive Whi5's repression of the heterodimeric transcriptional activators SBF and MBF [11]. This triggers expression of a large set of genes which includes the remaining two G_{1} cyclins *CLN1* and *CLN2* [12], resulting in a positive feedback loop that stabilizes once a critical threshold of Cln1/2 and hence Cdc28-Cln1/2/3 activity is reached [13]. High Cdc28-Cln1/2/3 activity directly or indirectly triggers polarization and bud emergence, as well as spindle pole duplication and DNA replication [3]. Mutations in the G_{1} network frequently lead to alterations in cell size at division [14], which is why the network has been proposed to impose size regulation, i.e. coordinate cell growth and division. However, the underlying mechanism driving the network is elusive and the role of proposed upstream components such as Whi3 and Ydj1 remains unclear [15]. Moreover, cell size distribution is known to be influenced by growth rate, ploidy and mutations in genes involved in growth, leading to the suggestion that ribosomal biogenesis or translation rate may determine the critical size threshold [14, 16, 17]. While this links the apparent size regulation of the cell division cycle to factors other than size *per se*, it has also been shown that additional constraints such as active size regulation are needed to provide for size homeostasis during autocatalytic growth [18].

The combinatorial regulation by distinct internal and external cues and the complex molecular machinery make the connection between cell size and the cell division cycle a prime target for mathematical modelling. The molecular network of the cell division cycle, its robustness and complex properties emerging from the network have been examined in several modelling efforts. These models range from detailed deterministic models of the regulatory circuitry to more abstract stochastic models [19-25]. However, most of these models directly define a critical size, a division ratio or both, making them unsuitable for studying the mechanisms underpinning the apparent size regulation *per se*. Furthermore, these models abstract growth to a steady exponential increase in cell volume and/or mass, and generally do not include metabolism at all. Conversely, most existing metabolic models are incompatible with cell division cycle modelling. For example, genome-wide metabolic models are too large to allow population level simulation and tools such as flux balance analysis are too abstract, as they do not account for costs and benefits of specific investments such as into ribosomes and metabolic enzymes [26-28]. In contrast, minimalistic models of basic metabolism in self-replicating systems such as those used to study growth rate related metabolic effects could potentially be integrated with a reduced cell division cycle model [29]. However, deterministic single cell models are not optimal for analysis of population properties such as size distributions. Characterization of populations has been addressed by population balance equations (PBEs) [30, 31] and cell ensemble modelling (reviewed in [32, 33]). PBEs derive population behaviour from analysis of single cell behaviour over time but require that the intracellular state can be described with a small set of variables. In contrast, ensemble modelling includes many individual cells that differ slightly, e.g. due to a stochastic component in the model. This is computationally more costly but allows a more detailed dynamic description, and hence ensemble modelling provides a framework in which to create an integrative model of metabolism and the cell division cycle, to allow the study of population effects. However, this has not yet been done and the current understanding of the mechanistic basis for size regulation builds on a critical size defined by the extensive G_{1} network [34] or the shuttling of critical cell cycle regulators between the growing cytoplasm and the slower growing nucleus [35].

Here, we re-examine the coupling of growth and division with a population level analysis of a model integrating metabolism and cell division. We designed this minimal core model to string together elementary concepts concerning cell cycle regulation and growth, including that cell cycle entry is limited by translation of a critical cell cycle regulator and that cell growth is autocatalytic and limited only by the nutrients in the growth medium [36]. This elementary design of the model also allows re-evaluation of the hypothesis that the G_{1} regulatory network is essential for cell size regulation. Surprisingly, our results indicate that autocatalytic growth is compatible with cell size homeostasis on the population level when considering geometric constraints on nutrient supply, and that it does not require more than a rudimentary version of the G_{1} regulatory network. While the full network has a strong effect on the actual size distribution in the cell population, size homeostasis itself can emerge from coupling between growth and division. We observe cell size homeostasis on the population level even when cells in the model lack any means to determine their sizes. Instead, size homeostasis emerges from growth due to the surface-area-to-volume constraint on metabolism. It is noteworthy that a very coarse grained model which only allows for the allocation of resources in two biomass pools – surface area and metabolic capacity – suffices for population level cell size regulation. Importantly, the model has only eight parameters, lacks regulatory circuitry, is stable over a wide range of growth rates and is robust against perturbations, after which the population quickly converges back to a stable attractor characteristic for its growth rate. However, the model fails to validate the hypothesis that cell cycle regulation occurs exclusively at the G_{1}/S boundary. Instead, it requires adaptation of both G_{1} and G_{2} duration to account for the empirically observed response to altered growth conditions. Despite this, it accurately describes a number of observations on the cellular level, such as (a) the difference in G_{1} duration between mother and daughter cells, (b) the cell size increase with genealogical age in the mother line and (c) the increase in average cell size with increasing growth rate. Taken together, our results show that cell size homeostasis can emerge from an elementary cellular design with fundamental physiological constraints and offer new insights into the complex coordination of growth and division.