A Model Linking Growth and Division

We employed a core model to assess the minimal requirement for size regulation in living cells (Fig. 2.3). This core model includes biomass production as a function of present biomass, i.e. a self-replicating system similar to previous work (Molenaar et al., 2009), and surface-to-volume ratio. Division occurs when the activity of a regulatory protein, the G₁ cyclin Cln1/2, reaches a given threshold. There is no regulatory feedback, size sensing, or size regulation. The model is based on two high level assumptions; (i) to grow, cells need to (a) take up nutrients and (b) incorporate nutrients in biomass and (ii) that metabolic efficiency decreases with decreasing area to volume ratio. The model accounts for two qualitatively different forms of biomass: structural biomass (B^A) and internal biomass (B^R). The structural biomass is proportional to the area (A), it includes cell wall and cell membrane and it determines the cellular uptake capacity for nutrients. The internal biomass is set proportional to the cellular metabolic capacity (R) and determines the ability to incorporate intracellular nutrients into new biomass.

The cell allocates resources to either structural or internal biomass depending on the cell cycle phase. Furthermore, structural biomass is allocated either to the mother (B^Am) or daughter (B^Ad) part of the cell and stays with that part, while internal biomass is split at division proportionally to volume.

The model includes a simplified cell cycle with only two phases: G₁, with growth of the unbudded mother and a bias towards allocation to internal biomass, and S/G₂, with polar growth into the bud, relatively small allocation to internal biomass and no growth of the mother. The phases are interspaced by START at which cells decide to divide -and an instantaneous M phase after a set time delay at which cells divide. All regulation occurs at the level of START, which is triggered by Cln1/2 accumulation in the nucleus.

METABOLISM

Figure 2.3:Modeling approach. (a) Cell cycle is approximated as two growth phases, G₁ and S/G₂, separated by two events: START and Mitosis. Growth allo-cates resources to two types of biomass: structural biomass (B^A) and internal biomass (B^R). B^Ais proportional to surface area (A), which determines vol-ume (V = A^3/2). B^R is proportional to metabolic (biosynthetic) capacity (R). Metabolism depends on uptake capacity (A) and relative metabolic capacity (R/V) and determines the resources available for growth. Thus, metabolism also determines translation ofmCLN1/2. Cln1/2 increase in G₁ triggers START transition. Regulation upstream of Cln1/2 was removed and CLN1/2 transcription is stochastic. (b) Resource allocation differs between two growth phases. In G₁ resources are allocated to mother cell and pri-marily toB^R. At START, cells polarize and start growing buds by targeted secretion, here described as an altered resource allocation. In S/G₂, resource allocation shifts towards B^A, specifically to the bud (B^Ad). S/G₂ duration is constant and Mitosis instantaneously separates mother and daughter cells.

Both retain their structural biomass (B^AmandB^Ad, respectively) and inherit a share of the internal biomass (B^R) proportional to their volume.

2.3 Results 1. nutrient uptake proportional to cell area

2. transcription is stochastic

3. biomass production is dependent on the internal biomass (B^R)

4. thus, all production reactions are implemented with second order kinetics, de-pendent on some precursor and the internal biomass (equations 2-5)

5. cells are approximated as spheres, thus V =A^3/2 6. cell area is sum of mother and daughter area 7. cell volume is sum of mother and daughter volume

8. cells may allocate their resources according to cell cycle stage to either structural or internal biomass

9. structural biomass can go into area mother (A^m) and area daughter (A^d), sep-arately

10. in G₁ there is no bud growth 11. after START only the bud grows

12. increase of metabolic capacity is strong in G₁ - less after START 13. threshold for nuclear kinase activity (zero order sensitivity)

14. there is targeted Cln1/2 destruction/nuclear exclusion after START 15. cells that are too old go quiet after∼ 24 divisions

16. S/G₂ is constant

17. mitotic cell division event is instantaneous

18. biomass precursors are always available (constant)

Table 2.3: List of modeling assumptions.

Importantly,CLN1/2 transcription is entirely stochastic to reflect the lack of upstream regulatory networks, see sections 1.2.1 and 2.2 for details. In summary, it comprises eight parameters (Tab. 2.1), seven ODEs, a function for stochastic transcription, six algebraic equations (Tab. 2.2) and rests on a set of explicit assumptions (Tab. 2.3).

2.3.2 The Model can Reproduce Characteristic Aspects of the Cell Cycle The core model was calibrated manually using high resolution literature data on cell growth that distinguishes between mother and bud growth (Aldea et al., 2007). Figures 2.4 and 2.5 show the fit as well as the trajectories of each of the five time-dependent vari-ables in a single cell over two cell division cycles, following only the mother in the second cycle. In this particular case, the newborn daughter spends a long time (∼100min) in her first G₁ phase, before the stochasticCLN1/2 expression leads to sufficient accumu-lation of Cln1/2 proteins to trigger START. In the S/G₂ phase, growth is redirected to the bud leading to an accelerated area and volume increase, which eventually leads to a biphasic growth pattern within one cycle. The phase specific alteration of the growth rate is in accordance with experimental observations (Aldea et al., 2007; Cookson et al., 2009; Goranov et al., 2009). Additionally, CLN1/2 transcription ceases and existing Cln1/2 proteins are actively degraded.

0 20 40 60 80 100 120 140 160 180 Time (min)

0 10 20 30 40 50 60 70

Mother +Bud

Mother

Bud Simulation whole cell

Simulation mother Simulation bud Experimental data

Cellvolume (fL)

Figure 2.4:The model captures single cell behavior. Model parameters were adjusted to in vivo growth data quantifying mother and daughter specific growth of budding cells (green dots; Aldea et al. (2007)). The adjusted pa-rameter values (Tab. 2.1) were used for all subsequent analysis. Solid lines indicate the simulation results.

2.3 Results After a set time delay, the cell passes through Mitosis and volume, area and metabolic capacity split between the mother and the daughter (resulting in a drop as only the mother line is followed). The cell enters her second G₁ larger and with higher metabolic capacity, resulting in a faster accumulation of the G₁ cyclins and hence a much shorter G₁ phase, which is in accordance with empirical data (Brewer et al., 1984; Cookson et al., 2009).

Figure 2.5: Single cell simulation over two cell cycles. Displayed are cell volume (upper panel), mCLN1/2 and Cln1/2 levels (middle panel) and structural and internal biomass (lower panel). Cell volume shows a biphasic growth pattern reflecting the shift in resource allocation from internal to structural biomass. CLN1/2 transcription is stochastic in G₁ but absent in S/G₂ and Cln1/2 is actively degraded during S/G₂. Note that the second G₁ phase is much shorter than the first.