1

𝛿 𝑥𝜎

𝑃 (𝛿 𝑥 )

0 1-1

Central limit theorem

Gaussians everywhere

Gaussians in physics

Slightly-disguised Gaussians in biology

𝑃 (𝑦𝑆𝑇 )

0𝑦 𝑆𝑇

𝛿 𝑦≅𝜕 (𝛿 𝑦 )𝜕 (𝛿𝑥1 )|𝐴𝑉𝐸 𝛿𝑥1+⋯

2

Central limit theorem

𝑃 (𝑥 )= 1𝜎 √2𝜋

𝑒− 12 ( 𝑥−𝜇𝜎 )

2

HT

𝛿 𝑥𝜎

𝑃 (𝛿 𝑥 )

0 1 2 3-1-2-3

3

𝛿 𝑥𝜎

𝑃 (𝛿 𝑥 )

0 1-1

Central limit theorem

Gaussians everywhere

Gaussians in physics

Slightly-disguised Gaussians in biology

𝑃 (𝑦𝑆𝑇 )

0𝑦 𝑆𝑇

𝛿 𝑦≅𝜕 (𝛿 𝑦 )𝜕 (𝛿𝑥1 )|𝐴𝑉𝐸 𝛿𝑥1+⋯

4

Physics lab: Engineered for tightly-controlled noise

5

Physics lab: Engineered for tightly-controlled noise

6

Physics lab: Engineered for tightly-controlled noise

t

V

x1

x2

x3

x5

x4

Hook vibration

Uneven air flow

Thermal expansion

Laser pointer vibration

Twisting

y

𝛿 𝑦=𝛿 𝑦 (𝛿𝑥1 , 𝛿𝑥2 , 𝛿𝑥3 ,⋯ )

𝛿 𝑦≅ 𝛿 𝑦 𝐴𝑉𝐸+𝜕 (𝛿 𝑦 )𝜕 (𝛿 𝑥1 )|𝐴𝑉𝐸𝛿𝑥1+ 𝜕 (𝛿 𝑦 )

𝜕 (𝛿𝑥2 )|𝐴𝑉𝐸𝛿𝑥2+⋯𝑌

𝑋 1 𝑋 2

“Small” noise: Neglect quadratic terms in Taylor expansion

0

7

𝛿 𝑥𝜎

𝑃 (𝛿 𝑥 )

0 1-1

Central limit theorem

Gaussians everywhere

Gaussians in physics

Slightly-disguised Gaussians in biology

𝑃 (𝑦𝑆𝑇 )

0𝑦 𝑆𝑇

𝛿 𝑦≅𝜕 (𝛿 𝑦 )𝜕 (𝛿𝑥1 )|𝐴𝑉𝐸 𝛿𝑥1+⋯

𝑑 𝑦𝑑𝑡

=𝜕 𝑦

𝜕𝑅+¿𝑑𝑅+¿

𝑑𝑡+ 𝜕 𝑦𝜕𝑅−

𝑑 𝑅−

𝑑𝑡¿¿

8

Biology: Law of mass action and logarithms

yx2x1 x3

y

y yy

y yy y y

𝑘+¿ ¿ 𝑘−𝑅+¿¿ 𝑅−

𝑘+¿ 𝑥1𝑥2𝑥3⋯ ¿ 𝑘−𝑦+1 -1

0=𝑘+¿ 𝑥1𝑥2 𝑥3⋯−𝑘− 𝑦𝑆𝑇 ¿

𝑘−𝑦 𝑆𝑇=𝑘+¿𝑥1𝑥2 𝑥3⋯ ¿

ln ( 𝑦𝑆𝑇 )=ln ¿¿Fluctuations in x1, x2, x3, etc. are not necessarily engineered to be small. First-order Taylor-expansion might be inaccurate.

ln ( 𝑦𝑆𝑇 )=ln ¿¿ln ( 𝑦𝑆𝑇 )− ln ¿¿

𝑌 𝑋 2𝑋 1

9

Biology: Law of mass action and logarithms

ln ( 𝑦𝑆𝑇 )− ln ¿¿𝑌 𝑋 2𝑋 1

A histogram of the logarithm of the concentration of y displays a normal distribution

30001507 3001500

𝑃 [ln ( 𝑦𝑆𝑇 ) ]

5 6 7 82 3 4ln ( 𝑦𝑆𝑇 )

𝑃 (𝑦𝑆𝑇 )

200 3001000 𝑦 𝑆𝑇

yST = e4 = 55

e5 = 150

e6 = 400