Mathematical Terms of Biological Information Theory

Thomas D. Schneider*

version = 1.11 of bitt.tex 2021 Jul 27

*National Institutes of Health, National Cancer Institute, Center for Cancer Research, RNA Biology Laboratory, P. O. Box B, Frederick, MD 21702-1201. (301) 846-5581; schneidt@mail.nih.gov; https://alum.mit.edu/www/toms/

B = log2M (bits) Number of bits for M symbols [1]
C = dspace log2(P∕N + 1) (bits/mmo)Molecular machine capacity [2, 3]
D = 2dspace ≪ 3n- 6 Dimensionality of a molecular machine coding space [2]
2Rln2- ln(ρ+1) ≤ D < 2Rln2 ϵr Lower and upper dimensionality bounds [4]
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]
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 [5, 6]
ϵt = ln(P+1) --NP--- N = ln(ρρ+1) Theoretical maximum molecular efficiency [5, 6, 4]
ϵr ≤ ϵt Real (or measured) molecular efficiency [5, 6]
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, 7]
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.
N (joules/mmo) Thermal noise flowing through a molecular machine during an opertion [2, 8, 9]
P = -ΔG (joules/mmo) Energy dissipated by a molecular machine in an opertion [2, 3]
p(x) = -√12πσ-2e-(x-2σμ2)2 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[10]
Renergy ≡-ΔG∘∕Emin (bits per mmo)The maximum bits that can be gained for the given free energy dissipation [5, 6]
ρ = P∕N 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, with an appendix on logarithms. Published on the web, 2013, 2013. https://doi.org/10.13140/2.1.2607.2000, https://alum.mit.edu/www/toms/papers/primer/.

[2]    T. D. Schneider. Theory of molecular machines. I. Channel capacity of molecular machines. J. Theor. Biol., 148:83–123, 1991. https://doi.org/10.1016/S0022-5193(05)80466-7, https://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. https://doi.org/10.1016/S0022-5193(05)80467-9, https://alum.mit.edu/www/toms/papers/edmm/.

[4]    T. D. Schneider and V. Jejjala. Restriction enzymes use a 24 dimensional coding space to recognize 6 base long DNA sequences. PLoS One, 14:e0222419, 2019. https://doi.org/10.1371/journal.pone.0222419, https://alum.mit.edu/www/toms/papers/lattice/.

[5]    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. https://doi.org/doi:10.1093/nar/gkq389, https://alum.mit.edu/www/toms/papers/emmgeo/.

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

[7]    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. https://doi.org/10.1088/0957-4484/5/1/001, https://alum.mit.edu/www/toms/papers/nano2/.

[8]    J. B. Johnson. Thermal agitation of electricity in conductors. Physical Review, 32:97–109, 1928. https://doi.org/10.1103/PhysRev.32.97.

[9]    H. Nyquist. Thermal agitation of electric charge in conductors. Physical Review, 32:110–113, 1928. https://doi.org/10.1103/PhysRev.32.110.

[10]    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. https://doi.org/10.1016/0022-2836(86)90165-8, https://alum.mit.edu/www/toms/papers/schneider1986/.