Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir

細菌體積的分布的變異係數 (CV) 能夠很好的維持在 0.1 以下。

doubling time model 出於實驗的觀察，假設細菌在兩 倍初始體積時對稱分裂，也就是 V d =2V b

doubling time model 根據實驗假設成長率為指數成長， 在時間 τ d 時體積為初始體積 V b 的兩 倍

doubling time model V d =f(V b ) 在時間 τ(V b ) 時，如果 f(V b ) 為 constant 則回到 critical size model constant τ d with 2V b =f(V b )

incremental model the cell attempts to add a constant volume v 0 to its newborn size

correlation coefficient

Variance Coefficient of Variation(CV)

Skewness

The model t a =appropriate amount of time t n =random noise time

assume t n to be Gaussian ∴ σ T 2 =var(t n )

f(V b ) 對平均出生體積 V avg 附近展開

Any two models that agree to lowest order will result in similar distributions We therefore choose to solve an equivalent model that will be amenable to analytic treatment corresponds

a=0 → f(V b ) =2V b t a = τ d doubling time model a=1 → f(V b ) =2V 0 critical size model a= 1/2 incremental model

next newborn volume Therefore

因為 t n 是 Gaussian variable ，我們 發現 也是 Gaussian variable 將 的變異數定為 σ v 2 → ∴ P(V b ) 將是 log-normal distribution

log-normal distribution 如果 X 是常態分佈的隨機變數， 則 exp(X) 為對數常態分佈分佈

log-normal distribution

the newborn size distribution

依據現實數據， σ v 只比 V 0 大數個 percent 。 類似的 size distribution

the newborn size distribution coefficient of variation skewness of the distribution is positive

the newborn size distribution

distribution of division times v b depends only on the noise t a is independent of t n and t n are Gaussian variables → the resulting interdivision time is tribution is also Gaussian

distribution of division times Variance given by α → 0 σ v diverges 只有在 0<a<2 有 stationary distribution

distribution of division times

correlation coefficient the correlation coefficient between mother size at birth and daughter size at birth is the same as that between size at birth and size at division.

X=V b y=V d For a narrow size distribution would not be very different than

斜率 V b 和 V b new 的斜率為 (1-a) V b 分布與 V b new 相同，因此 σ y = σ x ∴ V b new 和 V b 的相關係數為 0.5

Newborn Size: Generation time correlation coefficient

summary 1.Cell size depends exponentially on growth rate. 2.Cell size distributions are approximately log- normal. 3.The correlation coefficient between size at birth and division is approximately ½ 4.The correlation coefficient between size at birth and time to division is approximately -1/2

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