## Mathematical Terms of Biological Information Theory

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 [2, 3] D = 2dspace ≪ 3n- 6 Dimensionality of a molecular machine coding space [2] dspace = D∕2 Number of ‘pins’ a molecular machine uses [2, 3] ΔG (joules/mmo) Free energy dissipated by a molecular machine in an operation [2, 3] d.f. = 3n- 6 Degrees of freedom for n atoms [2] min = 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 [4, 5] ϵt = = Theoretical maximum molecular efficiency [4, 5] ϵr ≤ ϵt Real (or measured) molecular efficiency [4, 5] 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 [2, 6] 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 [2, 7, 8] Py = -ΔG (joules/mmo) Energy dissipated by a molecular machine in an opertion [2, 3] p(x) = e- 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 ≡-ΔG∘∕min (bits per mmo) The maximum bits that can be gained for the given free energy dissipation [4, 5] 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

[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/.

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