96
E.J. Brändas
Note that the term containing powers of the operator J in (4.36) inserted into (4.37)
translates to the sum
m−1
k=0
−it
τ
k 1
k!
≈ e
−it/τ
who’s magnitude is essentially one for large m and |t| < τ .
From Eq. (4.37) follows, dividing the lifetime τ into m/4π discrete time units,
or going from t − τ to t in single steps, defines a cumulative Poisson distribution
with the intensity (rate) parameter λ = 4π/m (or factoring out the triplet codon
λ = 4π/3m)
P
(t − τ )/τ < m/4π
=
m−1
k=0
4π
m
k 1
k!
e
−4π/m .
(4.38)
Equation (4.38) sustains the statistical response to the quantum-thermal chaos,
for each cell C i , via collective intra-cell quantum-thermal correlations leading up
to a cumulative statistical distribution. It is interesting to ponder whether the choice
m = |4π| = 12 carries any particular meaning, since it yields an almost critical
case with λ = 1.047, with P (0) = e −1.047 and P (1) = 1.047 × e −1.047 , where the
probability for one event of communication is larger than that for no event, being
exactly the same for λ = 1.
In summary we have mapped the basic constituents of the cell onto an STN configuration, i.e. spatio-temporally structured on the electron-proton transfer molecular level of the gene. Examining cell evolution and cell differentiation, based on
thermalization along with a degeneracy analysis, where the dimension of the largest
Jordan block, m, defines the so-called Segrè characteristic of the degenerate level,
one finds a particular type of statistics known as the (cumulative) Poisson distribution (4.31), (4.38). The number m ∝ ω 0 τ is essentially the cell’s Q-value factor,
cf. its use in regard to quality aspects of an oscillator or a resonator. One can e.g.
imagine the cell as a tuning fork coupled to a resonator or in the case of overdamping a “slamming door”. The factor 4π obtains naturally in the derivation and
may be given a geometric interpretation as being related to space angle integrations.
Note that standard practice usually gives the quality factor in terms of energy ratios
times 2π .
References
1. Eigen M (1931) Molecular self-organization and the early stages of evolution. Q Rev Biophys
4(2):149
2. Lehn J-M (2007) From supramolecular chemistry towards constitutional dynamic chemistry
and adaptive chemistry. Chem Soc Rev 36(2):151
3. Karsenti E (2007) Self-organization processes in living matter. Interdiscip Sci Rev 32(2):163
4. Michl J, Vacek J (1997) A molecular “Tinkertoy” construction kit: computer simulation of
molecular propellers. New J Chem 21(12):1259
5. Ladik J (1999) Polymers as solids: a quantum mechanical treatment. Phys Rep 313(4):171
E.J. Brändas
Note that the term containing powers of the operator J in (4.36) inserted into (4.37)
translates to the sum
m−1
k=0
−it
τ
k 1
k!
≈ e
−it/τ
who’s magnitude is essentially one for large m and |t| < τ .
From Eq. (4.37) follows, dividing the lifetime τ into m/4π discrete time units,
or going from t − τ to t in single steps, defines a cumulative Poisson distribution
with the intensity (rate) parameter λ = 4π/m (or factoring out the triplet codon
λ = 4π/3m)
P
(t − τ )/τ < m/4π
=
m−1
k=0
4π
m
k 1
k!
e
−4π/m .
(4.38)
Equation (4.38) sustains the statistical response to the quantum-thermal chaos,
for each cell C i , via collective intra-cell quantum-thermal correlations leading up
to a cumulative statistical distribution. It is interesting to ponder whether the choice
m = |4π| = 12 carries any particular meaning, since it yields an almost critical
case with λ = 1.047, with P (0) = e −1.047 and P (1) = 1.047 × e −1.047 , where the
probability for one event of communication is larger than that for no event, being
exactly the same for λ = 1.
In summary we have mapped the basic constituents of the cell onto an STN configuration, i.e. spatio-temporally structured on the electron-proton transfer molecular level of the gene. Examining cell evolution and cell differentiation, based on
thermalization along with a degeneracy analysis, where the dimension of the largest
Jordan block, m, defines the so-called Segrè characteristic of the degenerate level,
one finds a particular type of statistics known as the (cumulative) Poisson distribution (4.31), (4.38). The number m ∝ ω 0 τ is essentially the cell’s Q-value factor,
cf. its use in regard to quality aspects of an oscillator or a resonator. One can e.g.
imagine the cell as a tuning fork coupled to a resonator or in the case of overdamping a “slamming door”. The factor 4π obtains naturally in the derivation and
may be given a geometric interpretation as being related to space angle integrations.
Note that standard practice usually gives the quality factor in terms of energy ratios
times 2π .
References
1. Eigen M (1931) Molecular self-organization and the early stages of evolution. Q Rev Biophys
4(2):149
2. Lehn J-M (2007) From supramolecular chemistry towards constitutional dynamic chemistry
and adaptive chemistry. Chem Soc Rev 36(2):151
3. Karsenti E (2007) Self-organization processes in living matter. Interdiscip Sci Rev 32(2):163
4. Michl J, Vacek J (1997) A molecular “Tinkertoy” construction kit: computer simulation of
molecular propellers. New J Chem 21(12):1259
5. Ladik J (1999) Polymers as solids: a quantum mechanical treatment. Phys Rep 313(4):171