Mathematical Terms of Biological Information Theory

Thomas D. Schneider *

version = 1.08 of bitt.tex 2011 Nov 12

*National Institutes of Health, National Cancer Institute at Frederick, Gene Regulation and Chromosome Biology Laboratory, P. O. Box B, Frederick, MD 21702-1201. (301) 846-5581, email: schneidt@mail.nih.gov http://alum.mit.edu/www/toms/

B = log2M (bits) Number of bits for M symbols [1]



Cy = dspace log2(Py∕Ny + 1) (bits/mmo)Molecular machine capacity [23]



D = 2dspace 3n- 6 Dimensionality of a molecular machine coding space [2]



dspace = D∕2 Number of ‘pins’ a molecular machine uses [23]



ΔG (joules/mmo) Free energy dissipated by a molecular machine in an operation [23]



d.f. = 3n- 6 Degrees of freedom for n atoms [2]



Emin = kBT ln2 (joules per bit) A version of the Second Law of Thermodynamics that can be used as an ideal conversion factor between energy and bits [45]



ϵt =  (Py  )
ln-Ny+1-
  Py
  Ny = ln(ρ+1)
   ρ Theoretical maximum molecular efficiency [45]



ϵr ϵt Real (or measured) molecular efficiency [45]



kB (joules/kelvin) Boltzmann’s constant



λ = R∕2 Compressed bases: the number of bases a binding site would take up if the information of the site was compressed as small as possible.



M = 2B Number of symbols corresponding to B bits



mmo Molecular machine operation [26]



μ Mean of Gaussian distribution



σ Standard deviation of Gaussian distribution



π Circle circumference/radius, something to eat



n Number of atoms in a molecular machine. see d.f.



Ny (joules/mmo) Thermal noise flowing through a molecular machine during an opertion [278]



Py = -ΔG (joules/mmo) Energy dissipated by a molecular machine in an opertion [23]



p(x) = -√1---
  2πσ 2e-   2
(x-2σμ2) Probabilty of x for a Gaussian distribution
   



quincunx Galton’s Quincunx - a device that demonstrates the formation of a Gaussian distribution. See http://tinyurl.com/GaltonQuincunx



   
R (bits/mmo) Information gained during a molecular machine operation, often of a binding site[9]



Renergy ≡-ΔGEmin (bits per mmo)The maximum bits that can be gained for the given free energy dissipation [45]



ρ = Py∕Ny Energy dissipation of a molecular machine normalized by the thermal noise flowing through the machine



T (K) Absolute temperture, Kelvin



x Voltage (for a communications system) or total potental and kinetic energy for a molecular machine



y See x



References

References

[1]    T. D. Schneider. Information theory primer. Published on the web at http://alum.mit.edu/www/toms/papers/primer/, 2010.

[2]    T. D. Schneider. Theory of molecular machines. I. Channel capacity of molecular machines. J. Theor. Biol., 148:83–123, 1991. http://alum.mit.edu/www/toms/papers/ccmm/.

[3]    T. D. Schneider. Theory of molecular machines. II. Energy dissipation from molecular machines. J. Theor. Biol., 148:125–137, 1991. http://alum.mit.edu/www/toms/papers/edmm/.

[4]    T. D. Schneider. 70% efficiency of bistate molecular machines explained by information theory, high dimensional geometry and evolutionary convergence. Nucleic Acids Res., 38:5995–6006, 2010. doi:10.1093/nar/gkq389, http://alum.mit.edu/www/toms/papers/emmgeo/.

[5]    T. D. Schneider. A brief review of molecular information theory. Nano Communication Networks, 1:173–180, 2010. http://dx.doi.org/10.1016/j.nancom.2010.09.002, http://alum.mit.edu/www/toms/papers/brmit/.

[6]    T. D. Schneider. Sequence logos, machine/channel capacity, Maxwell’s demon, and molecular computers: a review of the theory of molecular machines. Nanotechnology, 5:1–18, 1994. http://alum.mit.edu/www/toms/papers/nano2/.

[7]    J. B. Johnson. Thermal agitation of electricity in conductors. Physical Review, 32:97–109, 1928.

[8]    H. Nyquist. Thermal agitation of electric charge in conductors. Physical Review, 32:110–113, 1928.

[9]    T. D. Schneider, G. D. Stormo, L. Gold, and A. Ehrenfeucht. Information content of binding sites on nucleotide sequences. J. Mol. Biol., 188:415–431, 1986. http://alum.mit.edu/www/toms/papers/schneider1986/.


U.S. Department of Health and Human Services  |  National Institutes of Health  |  National Cancer Institute  |  USA.gov  | 
Policies  |  Viewing Files  |  Accessibility  |  FOIA