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:
𝑑𝑉
𝑑𝑡 = ൜ 𝜆 linear growth
𝜆𝑉(𝑡) 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 – 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 𝑉௣௥௘ = 𝑉଴ +𝜆𝑇 or 𝑉௣௥௘ =𝑉଴𝑒ఒ். 𝑉଴ is the
initial cell volume, 𝜆 is the growth rate, and 𝑇 is the period of the cell cycle. 𝜆 does not depend
on 𝑉. One way the cell can ensure cell size stability is by regulating 𝑇 in response to sensing
different initial cell volumes 𝑉଴
. That is to say, the volume dependent growth response is 𝑔(𝑉)=𝜆𝑇(𝑉). Let 𝑓 be the volume partition factor. 𝑓=2 means the cell divides in half during
cytokinesis. For our model to accommodate asymmetric cell division 𝑓 can be ≠ 2. The function 𝜙(𝑉௡) relates the volume in the current generation to the volume in the next generation.
𝑉௡ାଵ =
𝑉௡ + 𝑔(𝑉௡)
𝑓 = 𝜙(𝑉௡) linear growth
𝑉௡ାଵ =
𝑉௡𝑒௚(௏೙)
𝑓 = 𝜙(𝑉௡) exponential growth
Let us further assume that there exists a fixed point at the average cell volume <𝑉>, i.e.
𝜙(< 𝑉 >) =< 𝑉 >, and that our cells do not deviate substantially from <𝑉>. In this case we
can approximate 𝜙(𝑉) near < 𝑉 > using a technique called Taylor-expansion.
𝑉௡ାଵ = 𝜙(𝑉௡
) ≈ 𝜙(< 𝑉 >) +
𝑑𝜙
𝑑𝑉௡
ฬழ௏வ
(𝑉௡−<𝑉 >)
𝑉௡ାଵ = 𝜙(𝑉௡
) ≈< 𝑉 > +
𝑑𝜙
𝑑𝑉௡
ฬழ௏வ
(𝑉௡−<𝑉 >)
𝑉௡ାଵ−< 𝑉 > =
𝑑𝜙
𝑑𝑉௡
ฬழ௏வ
(𝑉௡−< 𝑉 >)
The deviation from the average cell volume at the nth generation is ∆௡=𝑉௡−<𝑉>.
∆௡ାଵ =
𝑑𝜙
𝑑𝑉௡
ฬழ௏வ
∆௡
For the cell size to be stable across generations (not blow up to infinity or decrease to zero)
ฬ ௗథ
ௗ௏೙
ቚழ௏வ
ฬ < 1 (see pg. 356 in [2] for a simple proof). The red and green lines in the figure below are examples of the function 𝜙(𝑉௡
) 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: ௗథ
ௗ௏೙
ቚழ௏வ
= +1 𝑔(𝑉) = 𝑔ା(𝑉), ௗథ
ௗ௏೙
ቚழ௏வ
= −1 𝑔(𝑉) =𝑔ି(𝑉). By studying these limiting cases, we know all growth responses 𝑔(𝑉) that lead to stable cell
volumes lie between 𝑔_(𝑉) and 𝑔ା(𝑉). Either: 𝑔ି(𝑉) < 𝑔(𝑉) <𝑔ା(𝑉) or 𝑔ି(𝑉) >𝑔(𝑉)>𝑔ା(𝑉). If this criterion is met, size control takes place.
As 𝑔(𝑉) = 𝜆𝑇(𝑉), if we divide the inequalities by the growth rate 𝜆, we instead interpret this
bound as restricting possible durations for the cell cycle period in response the initial volume 𝑉to ensure cell size is stable across generations. This is profound. The cell is limited in how long it takes to divide.
𝑇ି(𝑉) < 𝑇(𝑉) < 𝑇ା(𝑉) or 𝑇ି(𝑉) > 𝑇(𝑉) > 𝑇ା(𝑉) for stable cell size
In the case of ௗథ
ௗ௏೙
ቚழ௏வ
= +1 no matter the initial volume, the cell volume repeats every
generation.
What is 𝑉௡ାଵ = 𝜙(𝑉௡
) in this case? (3 marks)
In the case of ௗథ
ௗ௏೙
ቚழ௏வ
= −1, no matter the initial volume, the cell volume repeats every two
generations. The cell volume alternates between < 𝑉 > +𝜖 and <𝑉 > −𝜖. What is 𝑉௡ାଵ = 𝜙(𝑉௡
) in this case? (5 marks)
For both the linear and exponential growth models, solve for 𝑔ା(𝑉௡) and 𝑔ି(𝑉௡) by substituting the equation for 𝜙(𝑉௡
) into 𝑉௡ାଵ =
௏೙ା௚(௏೙)
௙ = 𝜙(𝑉௡) 𝑉௡ାଵ =௏೙௘೒(ೇ೙)
௙ =𝜙(𝑉௡). (20 marks)
For both the linear and exponential model of growth, plot 𝑔(𝑉௡
) 𝑣𝑠 𝑉௡. In the plot shade the
regions where cell size is stable across multiple generations: i.e. between 𝑔_(𝑉) and 𝑔ା(𝑉). For growth responses outside these regions cell size either shrinks to 0 or blows up to infinity. Where do 𝑔_(𝑉) and 𝑔ା(𝑉) cross/intersect in the linear model? Where do 𝑔_(𝑉) and 𝑔ା(𝑉)
cross/intersect in the exponential model? Remember 𝑓 > 0. (5 marks)
1. H. Kubitschek, “Linear cell growth in escherichia coli