PoL I: Quantitative problems, dimensions, and units

Richard Neher
Biozentrum, University of Basel

How can physics help understand biological systems?

Mathematics is extremely effective in physics

• Motions of planets and moons can be predicted with extreme accuracy
• Behavior of atoms, particles, electrons etc can be controlled, predicted, and exploited
• Unintuitive behavior is captured by physical theories

Physics is simple

• Laws of physics connect fundamental particles and forces.
• Or describe homogenous matter.
• Biology is neither.
• Biological systems evolved -- layering complexity
• Cells consist of 10s of thousands of molecular species, interacting in complicated ways
• Biology is always in flux.

Much of biology describes complex dependencies in qualitative terms:

Wikipedia, by cybertory

Is there hope? YES!

Quantitative questions

• How rapidly does a protein move from one end of the cell to another?
• How many ribosomes/signaling molecules are in a cell?
• What fraction of transcription factors is bound to DNA?
• What are the speed limits for biochemical reactions?
• How are polarities and developmental gradients set up?
• How many human cells are in your body?
• How many other cells?
• Not: Gene X causes Y
• But: 20% faster diffusion of gene X extends the gradient by $20\mu m$

Dimensions and units

• length
• weight
• energy
• current
• force
• temperature
• ....
Units:
• length: meters, miles, feet, Angstrom
• weight: grams, stones,
• energy: Joules, calories
• current: ampere
• force: Newton
• temperature: Celsius, Kelvin
• ....
• Dimensions describe the nature of a quantity
• Units are conventions to measure them

Comparing quantities

• Only quantities of the same dimension can be compared: length $X$ is greater than length $Y$ etc. There is no sense is which length $X$ can be greater than weight $Y$.
• Units are conventions -- there is nothing fundamental about them.
• Some unit systems are more convenient than others for a particular purpose.
• Everyday units are often inconvienent for biological processes
• We can pick units to make things as simple as possible

Laws of physics relate quantities of different dimensions

Take Newton's law:
$$F = m \times a$$ $$\mathrm{Force} = \mathrm{mass} \times \mathrm{acceleration}$$
This equation relates quantities of different dimensions to each other.
In our conventional unit system the different quantities have the units:
$$\mathrm{[Newton]} = \mathrm{[Kilogram]} \times \mathrm{[meter/second^2]}$$
thereby defining the unit of force "Newton".

The law is valid in any system of units!
Units are chosen by humans, the law itself is fundamental.

Physical constants

Consider the gravitational law:
$$F \sim \frac{m_1 m_2}{r^2}$$ $$\mathrm{Force} \sim \mathrm{mass^2}/\mathrm{length^2}$$
The quantities left and right do not have the same dimensions! This relation tells us how the force changes when masses or distance change, but not what the absolute value of the force is.
The missing link is established by the gravitational constant $G$:
$$F = G\frac{m_1 m_2}{r^2}\quad \mathrm{with} \quad G \approx 6.67 \frac{m^3}{kg\,s}$$
The value of $G$ depends on the system of units we use.

This course: quantitative understanding of biology

• Laws of physics yield quantitative and predictive relationships and constraints for biology
• Computer programming is essential to explore quantitative relationships and handle data
• Units are absolutely critical to get right:
If you calculate a length and the result is in seconds something went seriously wrong
• Checking your units will catch many obvious mistakes.
• Choosing your units wisely will make many calculations easier