# models for cell growth

## models for cell growth

Q2 (33pts)

In this question we model the volume of a cell as it grows and divides. It is extremely important for the cell size post division to be stable across generations. Two simple models for cell growth are:

๐๐

๐๐ก = เต ๐ linear growth

๐๐(๐ก) exponential growth

In the first case, cell volume increases linearly with time. This is hypothesized to be a result of

growth-limiting factor such as nutrient import which is independent of cell size [1]. In the second case, the growth rate is proportional to cell size โ assumed to be a consequence from larger cells having a proportionally increased capacity for metabolism compared to smaller cells.

The volume of the cell just prior to division is then ๐เฏฃเฏฅเฏ = ๐เฌด +๐๐ or ๐เฏฃเฏฅเฏ =๐เฌด๐เฐเฏ. ๐เฌด is the

initial cell volume, ๐ is the growth rate, and ๐ is the period of the cell cycle. ๐ does not depend

on ๐. One way the cell can ensure cell size stability is by regulating ๐ in response to sensing

different initial cell volumes ๐เฌด

. That is to say, the volume dependent growth response is ๐(๐)=๐๐(๐). Let ๐ be the volume partition factor. ๐=2 means the cell divides in half during

cytokinesis. For our model to accommodate asymmetric cell division ๐ can be โ 2. The function ๐(๐เฏก) relates the volume in the current generation to the volume in the next generation.

๐เฏกเฌพเฌต =

๐เฏก + ๐(๐เฏก)

๐ = ๐(๐เฏก) linear growth

๐เฏกเฌพเฌต =

๐เฏก๐เฏ(เฏเณ)

๐ = ๐(๐เฏก) exponential growth

Let us further assume that there exists a fixed point at the average cell volume <๐>, i.e.

๐(< ๐ >) =< ๐ >, and that our cells do not deviate substantially from <๐>. In this case we

can approximate ๐(๐) near < ๐ > using a technique called Taylor-expansion.

๐เฏกเฌพเฌต = ๐(๐เฏก

) โ ๐(< ๐ >) +

๐๐

๐๐เฏก

เธฌเฎดเฏเฎต

(๐เฏกโ<๐ >)

๐เฏกเฌพเฌต = ๐(๐เฏก

) โ< ๐ > +

๐๐

๐๐เฏก

เธฌเฎดเฏเฎต

(๐เฏกโ<๐ >)

๐เฏกเฌพเฌตโ< ๐ > =

๐๐

๐๐เฏก

เธฌเฎดเฏเฎต

(๐เฏกโ< ๐ >)

The deviation from the average cell volume at the nth generation is โเฏก=๐เฏกโ<๐>.

โเฏกเฌพเฌต =

๐๐

๐๐เฏก

เธฌเฎดเฏเฎต

โเฏก

For the cell size to be stable across generations (not blow up to infinity or decrease to zero)

เธฌ เฏเฐฅ

เฏเฏเณ

แเฎดเฏเฎต

เธฌ < 1 (see pg. 356 in [2] for a simple proof). The red and green lines in the figure below are examples of the function ๐(๐เฏก

) which result in cell size converging or diverging over

successive generations. Red trajectories are divergent and green trajectories are convergent.

We are interested in determining the boundary of the growth response where stability breaks

down: เฏเฐฅ

เฏเฏเณ

แเฎดเฏเฎต

= +1 ๐(๐) = ๐เฌพ(๐), เฏเฐฅ

เฏเฏเณ

แเฎดเฏเฎต

= โ1 ๐(๐) =๐เฌฟ(๐). By studying these limiting cases, we know all growth responses ๐(๐) that lead to stable cell

volumes lie between ๐_(๐) and ๐เฌพ(๐). Either: ๐เฌฟ(๐) < ๐(๐) <๐เฌพ(๐) or ๐เฌฟ(๐) >๐(๐)>๐เฌพ(๐). If this criterion is met, size control takes place.

As ๐(๐) = ๐๐(๐), if we divide the inequalities by the growth rate ๐, we instead interpret this

bound as restricting possible durations for the cell cycle period in response the initial volume ๐to ensure cell size is stable across generations. This is profound. The cell is limited in how long it takes to divide.

๐เฌฟ(๐) < ๐(๐) < ๐เฌพ(๐) or ๐เฌฟ(๐) > ๐(๐) > ๐เฌพ(๐) for stable cell size

In the case of เฏเฐฅ

เฏเฏเณ

แเฎดเฏเฎต

= +1 no matter the initial volume, the cell volume repeats every

generation.

What is ๐เฏกเฌพเฌต = ๐(๐เฏก

) in this case? (3 marks)

In the case of เฏเฐฅ

เฏเฏเณ

แเฎดเฏเฎต

= โ1, no matter the initial volume, the cell volume repeats every two

generations. The cell volume alternates between < ๐ > +๐ and <๐ > โ๐. What is ๐เฏกเฌพเฌต = ๐(๐เฏก

) in this case? (5 marks)

For both the linear and exponential growth models, solve for ๐เฌพ(๐เฏก) and ๐เฌฟ(๐เฏก) by substituting the equation for ๐(๐เฏก

) into ๐เฏกเฌพเฌต =

เฏเณเฌพเฏ(เฏเณ)

เฏ = ๐(๐เฏก) ๐เฏกเฌพเฌต =เฏเณเฏเณ(เณเณ)

เฏ =๐(๐เฏก). (20 marks)

For both the linear and exponential model of growth, plot ๐(๐เฏก

) ๐ฃ๐ ๐เฏก. In the plot shade the

regions where cell size is stable across multiple generations: i.e. between ๐_(๐) and ๐เฌพ(๐). For growth responses outside these regions cell size either shrinks to 0 or blows up to infinity. Where do ๐_(๐) and ๐เฌพ(๐) cross/intersect in the linear model? Where do ๐_(๐) and ๐เฌพ(๐)

cross/intersect in the exponential model? Remember ๐ > 0. (5 marks)

1. H. Kubitschek, โLinear cell growth in escherichia coli

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