Cell Size Regulation in BacteriaDepartment 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

Vd=2Vb

doubling time model• 根據實驗假設成長率為指數成長，

在時間 τd 時體積為初始體積 Vb 的兩倍

doubling time modelVd=f(Vb) 在時間 τ(Vb) 時，如果 f(Vb)為 constant 則回到 critical size model

constant τd with 2Vb =f(Vb)

incremental model• the cell attempts to add a constant

volume v0 to its newborn size

correlation coefficient

Variance

Coefficient of Variation(CV)

Skewness

The model

ta=appropriate amount of time

tn=random noise time

• assume tn to be Gaussian

∴σT

2=var(tn)

• f(Vb) 對平均出生體積 Vavg 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(Vb) =2Vb ta= τd

doubling time model• a=1 → f(Vb) =2V0

critical size model

• a=1/2

incremental model

• next newborn volume

Therefore

• 因為 tn 是 Gaussian variable ，我們發現 也是 Gaussian variable

• 將 的變異數定為 σv2

→∴P(Vb)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

the newborn size distribution

• 依據現實數據， σv 只比 V0 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

• vb depends only on the noise ta is independent of tn

• and tn are Gaussian variables

→ the resulting interdivision time is tribution is also Gaussian

distribution of division times• Variance given by

• α → 0 σv 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=Vb y=Vd

• For a narrow size distribution would not be very different than

• 斜率Vb 和 Vb

new 的斜率為 (1-a)

• Vb 分布與 Vbnew 相同，因此 σy = σx

∴Vbnew

和 Vb 的相關係數為 0.5

Newborn Size: Generation time correlation coefficient

summary1. 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 ½4. The correlation coefficient between size at

birth and time to division is approximately -1/2

END