Physical Principles in BiologyBiology 3550

Fall 2018

Lecture 4:

A Bit More on Measurement and Unitsand

Introduction to Probability

Monday, 27 August 2018

c©David P. GoldenbergUniversity of Utah

goldenberg@biology.utah.edu

Announcements

First Quiz: Friday, 31 Aug.25 min, second half of class.

First problem setPosted on class web page; due Tuesday, 4 Sept.

Discussion Sessions.

• Tuesdays, 11:00 AM – noon. Life Science 102

• Wednesdays, 8:30 – 9:30 AM. Gardner Common4680

Office hours:• Tuesdays: 9:30 - 11:00 AM

• Wednesdays: 2:30 - 3:30 PM

• Other times by appointment.Send me an e-mail message!

Clicker Question from Last Time

How many hydrogen ions (H+) are within a typical bacterium?

1 1

2 100

3 1 thousand

4 1 million (106)

5 1 billion (109)

All answers count (for now)!

How Many Hydrogen Ions Are in a Typical Bacterium:

Poll Results

Scale and Dimensions of a Bacterial Cell

A typical bacterium found in the human gut: Escherichia coli

Approximate this as a cylinder

Volume of cylinder = L × AL = length of cylinder

A = area of cap = π × R2

Estimated volume: 1.6×10−15 L

Units of Concentration

Most convenient: amount of solute per volume of solution• g/L (= mg/mL): 1 g solute in 1 L final volume of solution

• molar (M) = mole/L: 1 mole of solute in 1 L final volume of solution

1 mole = amount of a substance containing the number of atoms ormolecules equal to the number of atoms in 12 g of 12C.

Number of atoms or molecules in 1 mole of a substance is calledAvogadro’s number, NA ≈ 6.02×10

23

Some less convenient (for purposes of calculation) units ofconcentration• molal: 1 mole of solute dissolved in 1 kg solvent

• 1%(m/v): 1 g solute in 100 mL final volume of solution

• 1%(v/v): 1 mL pure liquid in 100 mL final volume of solution

A Source of Confusion: Units for “Molecular Weight”

Molecular weight or molecular mass:• The mass of a single molecule

• Units: atomic mass unit (u or amu) or dalton (Da) or kilodalton (kDa)Defined by saying that the mass of 1 atom of 12C is exactly 12 amu or12 Da.

1 amu = 1 Da = mass of one atom of 12C ÷12• Units are often not included, because it is really a relative mass, Mr.

• amu is commonly used in mass spectrometry

• Da and kDa are very commonly used in biochemistry and molecularbiology, especially for proteins and other macromolecules.

Molar mass:• Mass of one mole of a compound

• Units: g/mol

Molecular mass of 100 Da→ molar mass of 100 g/mol

To Calculate the Amount of Solute in a Solution

The number of grams in 53 mL of a 5 g/L solution:

53mL × 0.001 L/mL = 0.053 L0.053 L × 5 g/L = 0.26 g

The number of moles in 1.3 L of a 15 mM solution (1mM = 0.001M):

15mM × 0.001M/mM = 0.015M = 0.015mol/L1.3 L × 0.015mol/L = 0.0195mol

The number of molecules in 1.3 L of a 15 mM solution:

1mol = 6.02×1023molecules0.0195mol × 6.02×1023molecules/mol = 1.17×1022molecules

Clicker Question #1

How many moles of water molecules (Mr = 18) are in 1 L?

A) ∼ 10

B) ∼ 30

C) ∼ 50

D) ∼ 70

1000 g ÷ 18 g/mol = 56mol

A Special Measure of Concentration for Hydrogen Ions

Dissociation of water

H2O −−−→←−−− H+ + OH–

Hydrogen ion concentration expressed as pH

pH = − log [H+]

with [H+] expressed in molar units

To convert from pH to molar concentration:

[H+] = 10−pHM

In a neutral solution, [H+] = [OH−]

This happens when pH= 7.

How Many H+ Ions Are There in a Bacterium?

Volume = 1.6×10−15 L

[H+] = 10−pHM = 10−7M

Moles of H+:

1.6×10−15 L × 10−7mol/L = 1.6×10−22moles

Number of ions:

1.6×10−22moles × 6.02×1023 ions/mol ≈ 100H+ ions

Some bacteria grow at pH 9. How many hydrogen ions are in one ofthese bacteria?

Direction Change

Warning!

Probability

Random Motion of Latex Beads in Water

Brownian Motion Movie

What makes them move?

YouTube movie: https://www.youtube.com/watch?v=cDcprgWiQEY

BrownianMotion.mp4Media File (video/mp4)

https://www.youtube.com/watch?v=cDcprgWiQEY

Robert Brown

1773-1858

Scottish botanist

In 1827, observed random motions of small (1µm) particles withinpollen grains.

Are they alive?

Albert Einstein

1879-1955 (photo 1904)

1905: Einstein’s Annus MirabilisSpecial relativity

E = mc2

Photoelectric effect

Brownian motion

Photograph by Lucien Chavan, https://en.wikipedia.org/wiki/Albert_Einstein

https://en.wikipedia.org/wiki/Albert_Einstein

Simulation of Brownian Motion

https://en.wikipedia.org/wiki/Brownian_motion

http://weelookang.blogspot.com/2010/06/ejs-open-source-brownian-motion-gas.html

A realistic molecular simulation is very difficult!

https://en.wikipedia.org/wiki/Brownian_motionhttp://weelookang.blogspot.com/2010/06/ejs-open-source-brownian-motion-gas.html

A Random Walk in One Dimension

Heads - East Tails - West

Results of the Coin Toss Experiment

T H T T H H H H H T H H T T T H H T H T 11

H T T H T H H T T T T H T T H T H H T H 9

H T T H T H T T H T H H T H T H T H T T 9

T H H T H T H H H T H T T T T T H T H T 9

T T H H T H H H H H H T H T T H T T H T 11

H T T T H H H H H T H H H T H T T H H H 13

H T T H T H H H T T H T T H H H H H T H 12

T H H H T T H T H T T T T T T T T T H H 7

H H H H T H H T T T T T H T T H T H T T 9

H H H T H T H H H H H T H H T T H T H T 13

H T H T T T H T T H H H T H T H H T H H 11

H H T T T T T T T H H H T H H T H H T T 9

Results of the Coin Toss Experiment

H T H T T H T H T T T H T T H H T T T H 8

T T T H H H T H T H H T T T H T T T T H 8

T T T T H T H H H T T H H H H H T T H H 11

H H H T H H T T H H H H H H H T T T H H 14

H H H T H T T H T T H H H H T T T T T T 9

T T H H T H H T T H H T T H T H T T H T 9

T H T H H T T T T H T H H H H T H H T H 11

H T T H H H T H H T T T H T T H H T H T 10

H H T H T T H H H H T T T H T H H T H T 11

T T H T T T T T H H H T H H T H T H H T 9

H T H H T H T T H T H T H T H H H T T H 11

H H T T T T T T T H T H H T H T H H T H 9

Some Statistics from the Coin Toss Experiment

Total sequences: 24

Total coin tosses: 480

Total heads: 243

Total tails: 237

Fraction heads: 0.51

Fraction tails: 0.49

Two Kinds of Distances in a Random Walk

Heads - East Tails - West

Total distance traveled: Pedometer distance.The number of steps times the length of each step.

The final displacement from the starting point.

Displacement from the Starting Point

for a One-Dimensional Random Walk

Heads - rightTails - left

Start at position x = 0.

Take n random steps to the right orleft.

nH = no. of heads

nT = no. of tails

Final position is x .x = nH − nT

(in units of step lengths)

Generally expect a distribution of x ifthe random walk is repeated a largenumber of times.

Distribution of Displacements from Our Coin Tosses

What patterns can we find?

We need more data!

Simulation of a Random Walk: The Galton Probability Machine

Sir Francis Galton (1822-1911)• Cousin of Charles Darwin

• Applied statistical and probabilistic reasoning to biological variation.

• Early advocate of eugenics, improvement of human species by breeding.

http://mathworld.wolfram.com/GaltonBoard.html

https://en.wikipedia.org/wiki/Francis_Galton

http://mathworld.wolfram.com/GaltonBoard.htmlhttps://en.wikipedia.org/wiki/Francis_Galton

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