Molecular Information Theory

1. Molecular Information Theory Niru Chennagiri Probability and Statistics Fall 2004 Dr. Michael Partensky

2. Overview • Why do we study Molecular Info. Theory? • What are molecular machines? • Power of Logarithm • Components of a Communication System • Discrete Noiseless System • Channel Capacity • Molecular Machine Capacity

3. Motivation • Needle in a haystack situation. • How will you go about looking for the needle? • How much energy you need to spend? • How fast can you find the needle? • Haystack = DNA, Needle = Binding site, You = Ribosome

4. What is a Molecular Machine? • One or more molecules or a molecular complex: not a macroscopic reaction. • Performs a specific function. • Energized before the reaction. • Dissipates energy during reaction. • Gains information. • An isothermal engine.

5. Where is the candy? • Is it in the left four boxes? • Is it in the bottom four boxes? • Is it in the front four boxes? You need answer to three questions to find the candy Box labels: 000, 001, 010, 011, 100, 101, 110, 111 Need log8 = 3 bits of information

6. More candies… • Box labels: 00, 01, 10, 11, 00, 01, 10, 11 • Candy in both boxes labeled 01. • Need only log8 - log2 = 2 bits of information. In general, m boxes with n candies need log m - log n bits of information

7. Ribosomes 2600 binding sites from 4.7 million base pairs Need log(4.7 million) - log(2600) = 10.8 bits of information.

8. Communication System

9. Information Source • Represented by a stochastic process • Mathematically a Markov chain • We are interested in ergodic sources: Every sequence is statistically same as every other sequence.

10. How much information is produced? Measure of uncertainty H should be: • Continuous in the probability. • Monotonic increasing function of the number of events. • When a choice is broken down into two successive choices, Total H = weighted sum of individual H

11. Enter Entropy

12. Properties of Entropy • H is zero iff all but one p are zero. • H is never negative. • H is maximum when all the events are equally probable • If x and y are two events H(x,y)£ H(x) + H(y) • Conditional entropy: Hx(y)£ H(y)

13. Why is entropy important? • Entropy is a measure of uncertainty. • Entropy relation from thermodynamics • Also from thermodynamics • For every bit of information gained, the machine dissipates kBTln2 joules.

14. Ribosome binding sites

15. Information in sequence

16. Information curve Information gain for site l is Plot of this across the sites gives Information curve. For E.Coli, Total information is about 11 bits. … same as what the ribosome needs.

17. Sequence Logo

18. Channel capacity Source transmitting 0 and 1 at 1000 symbols/sec. 1 in 100 symbols have an error. What is the rate of transmission? Need to apply a correction correction = uncertainty in x for a given value of y Same as conditional entropy = 81 bits/sec

19. Channel capacity contd. For a continuous source with white noise, Signal to noise ratio Bandwidth Shannon’s theorem: As long as the rate of transmission is below C, the number of errors can me made as small as needed.

20. Molecular Machine Capacity • Lock and key mechanism. • Each pin on the ribosome is a simple harmonic oscillator in thermal bath. • Velocity of the pins represented by points in 2-d velocity space • More pins -> more dimensions. • Distribution of points is spherical.

21. Machine capacity For larger dimensions: All points are in a thin spherical shell Radius of the shell is the velocity and hence square root of the energy Before binding: After Binding:

22. Number of choices = Number of ‘after’ spheres that can sit in the ‘before’ sphere =Vol. of Before sphere/Vol. Of after sphere Machine capacity = logarithm of number of choices

23. References