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

Dec 17, 2015

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Page 1: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

Cell Size Regulation in BacteriaDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Ariel Amir

Page 2: 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。

Page 3: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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

Page 4: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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

Page 5: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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)

Page 6: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

incremental model• the cell attempts to add a constant

volume v0 to its newborn size

Page 7: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

correlation coefficient

Page 8: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

Variance

Coefficient of Variation(CV)

Page 9: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

Skewness

Page 10: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

The model

ta=appropriate amount of time

tn=random noise time

Page 11: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• assume tn to be Gaussian

∴σT

2=var(tn)

Page 12: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• f(Vb) 對平均出生體積 Vavg 附近展開

Page 13: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• 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

Page 14: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• a=0 → f(Vb) =2Vb ta= τd

doubling time model• a=1 → f(Vb) =2V0

critical size model

• a=1/2

incremental model

Page 15: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• next newborn volume

Therefore

Page 16: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• 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

Page 17: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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

Page 18: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

log-normal distribution

Page 19: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

the newborn size distribution

Page 20: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

the newborn size distribution

• 依據現實數據, σv 只比 V0 大數個percent 。

• 類似的 size distribution

Page 21: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

the newborn size distribution• coefficient of variation

• skewness of the distribution is positive

Page 22: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

the newborn size distribution

Page 23: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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

Page 24: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

distribution of division times• Variance given by

• α → 0 σv diverges

• 只有在 0<a<2 有 stationary distribution

Page 25: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

distribution of division times

Page 26: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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.

Page 27: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• X=Vb y=Vd

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

Page 28: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

• 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

Page 29: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

Newborn Size: Generation time correlation coefficient

Page 30: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

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

Page 31: Cell Size Regulation in Bacteria Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Ariel Amir.

END