lecture2_statistics and radom walk and diffusion_updated on feb2nd

25%-quiz 1

25% quiz 2

50% final exam

Grading Format

~8 MCQs

~4 Assays with calculations

Open book

Agenda • Jan 16 --- Lecture 1 ( Cell, bacterial, and biomolecule) • Jan 23 --- Lecture 2 (statistics and Diffusion) • Jan 30 --- Break (Chinese new year) • Feb 6 --- Lecture 3 (Thermodynamics and Chemical reaction) • Feb 13 --- No class • Feb 20 --- Lecture 4 (Characterization technologies) • Feb 27 --- Exam I (open book) • Mar 6 --- Recess week • Mar 13 --- Guest lecture 1 (optical imaging strategies) Dr. Audrey Shi

(Leica) • Mar 20 --- Guest lecture 2 (MRI for biology/physiology) Prof. ShaoYing

Huang (SUTD) • Mar 27 --- Guest lecture 3 (cell-cell/substrate interaction) Dr. Nils Gauthier

(NUS) • Apr 3 --- Guest lecture 4 (development of electron microscope for studying

plasmid, protein, and cell) Prof. Utkur Mirsaidov (NUS) • Apr 10 – Exam II (open book) • Apr 17 – Overview

We are mostly made of

water (H2O): ≈75%

C —very versatile: everything made of: Nucleic acids,

Proteins, Lipids (fat), Carbohydrates.

O —bonding, proteins, fats, nucleic acids

N — proteins, genetic material

Ca, P — bones

What are we made of?

Biophysics • Biophysics is a bridge between biology and

physics

Biophysics Applications

• Biomolecular interaction analysis

• membrane protein and lipid studies

• Drug discovery and development (protein-small molecular interaction)

• Protein-metal ion interaction

• Biomolecular structure and stability

• Protein crystallography studies

Two areas of BioPhysics

Method Development

New techniques to see better Model Development

What it means (lectures 2-3)

Model Development

• Statistics

• Random Walks, Friction and Diffusion

• Thermodynamics

• Chemical reaction and biological machines

Statistics

Probability in Science

You can NOT tell the exact state of each gas molecular in the room; BUT you can measure the precise probability distribution of gas molecules

0.259X60=

• For the babies distribution, find the variance of x

What is the probability of getting a 6 with a dice plus getting head with a coin? What is the probability of getting a 6 with a dice plus getting a 1 with another dice?

Integral of P(r) from R0 to infinite

Low-density gases obey a universal law

• The number of gas molecules in a box at atmospheric pressure is proportional to its volume. (V↔N)

• Further, p*V=N*kB*T (Ideal gas law) 1. kB is Boltzmann constant and is 1.38*E(-21) joules

per degree

2. T is absolute temperature. (t+273C)

3. At room temperature (22C), kB*T equals 4.1E(-21)J or 4.1pN*nm

What about a dirt particle with 1.25E(-10)kg?

Compared mgh and 3/2KT

Simple case: Ball in gravitational field.

Thermal fluctuations, finite probability of being at height, h.

E = ??

Partition Function for 2-state system

1

o 1

1

o

-E /kT

1 -E /kT -E /kT

- 1 E

- o E

eP E =

e +e

o

1

E = mgh

E h = 0

E h =(mg)(h meter)

P(h) =

P(0)

As ball gets smaller, probability gets smaller / larger ?

“Ball” the size of O2? Why can you breathe standing up?

For O2, 1/e height is ~10 km ~height of Mt. Everest.

(10 kM is “death zone”)

What is 1/e height for O2?

-mgh/kT e

Random Walks, Friction and Diffusion

http://youtu.be/STLAJH7_zkY

1. Why does the dye spread through the solution?

2. Why does the diffusion take a random/unpredicted path?

3. Why is the diffusion quicker at higher temperature?

http://youtu.be/cDcprgWiQEY

Probability for getting back to the starting point

L=1; R=-1 To return to the starting point, the head has to be same as the tail

P=M0/M M0=possible combinations M=total way of flip coming out

HT or TH

HHTT or TTHH HTHT or THTH THHT or HTTH

http://www.math-prof.com/Prob/Prob_Ch_05.asp https://www.khanacademy.org/math/probability/probability-and-combinatorics-topic/probability_combinatorics/v/probability-using-combinations

Probability using Combinations

• Total number of way all the flip can come out

• 5000 heads out of 10000

The exclamation point denotes the Factorial function

Factorial function is a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n

4!=4X3X2X1=24

0!=1 N!=(n-1)!Xn

Displacement and diffusion law

RRR RRL RLR LRR LRL LLR RLL LLL

When L=1cm

The diffusion law is model independent

Key: image the bin to be very narrow; the difference between a function, line N(x), at two nearby points is L times the derivative of N

Key: the development of the density won’t depend on how big the box is; the important thing is the number per unit area of a

1. J measures the net number of particles moving from left to right;

2. A net drift to the right ensures, tending to even out the distribution, or make it more uniform.

3. If there is structure in the original distribution, Fick’s law says that diffusion will tend to erase it.

4. Diffusion constant D enters the formula because more-rapidly diffusing particles will erase their order fast

B: partition coefficient of solute in oil versus water D: diffusion constant in oil

Constant I=J*4πr^2

More than diffusion

+

+ +

+ + +

?

112

http://youtu.be/Md0PtdRxXvw http://youtu.be/Ya-jpBgzcc8

113

C(z): profile of particles density

Adapted from Nernst relation

114

Scale height

115

1g/ml

Not a stable suspension

116

Scale height also depends on the acceleration constant

117

Comparable with z and cause a significant difference between Cx and C0

118

Sedimentation equilibrium in centrifuge

119

Or sedimentation coefficient

-1

49

120

Log-log plot of diffusion constant D of polymethyl methacrylate in acetone as a function of the polymer’s molar mass M.

P=0.57

121

Log-log plot of sedimentation coefficients of polymethyl methacrylate in acetone as a function of the polymer’s molar mass M.

122

http://youtu.be/_dbnH-BBSNo

123

Definition of laminar flow

124

Fixed

Pushed to right

125

f

Any fluid obeying is called a Newtonian fluid.

Characteristics of Newtonian Fluid

1. Obey ;

2. Completely characterized by its viscosity and mass density;

3. Isotropic (the same in every direction)

126

𝑓 = −𝜂𝑣0𝐴

𝑑

127

f

𝑓 ∗ 𝜌𝑚𝞰2

128

129

http://youtu.be/fma5WBl90Sw

130 Now we want to compare ftotal with ffrict

If one is much bigger than another, we could drop the smaller on in Newton law

131

132

133

134

135

136

137

138

139

140

Calculate travel distance

141

142

Cilia motion is periodic but not reciprocal

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