The name is absent

A.P. Kirilyuk

exponentially with the number of elements, contrary to power-law dependence in usual,
dynamically single-valued models (section 3). Being applied to various important cases
of interaction development in living organisms, such as genome dynamics or brain
operation, this result explains their huge, qualitative advantages with respect to any
conventional simulation that underlie all the “miracles of life” (self-reproduction,
adaptable evolution, intelligence, consciousness, etc.). Important practical conclusions
for genetic research strategy are derived from the unreduced fractal structure of genome
interaction dynamics (section 4). In that way we substantiate and specify the necessary
change in life sciences and related fields, which can uniquely solve the growing
“difficult” (e.g. “ethical”) problems of the modern blind, purely empirical technology
development and provide the basis for the truly sustainable future. The latter involves
genuine, causally complete understanding and control of living form emergence and
dynamics, at any level of interest, giving rise to new possibilities in both fundamental
(e.g. mathematical) and applied aspects of knowledge, including such directions as
constructive genetics and integral medicine [9,10].

2 Probabilistic fractal structure of a generic interaction process

We start from interaction problem between arbitrary (but known) system components,
such as brain neurons, cell elements, or genes. It can be expressed by the existence
equation that generalises many particular, model dynamic equations [9-13]:

∙∑ hk(qk) + ∑v«(qk,q∣) Ψ(Q) = eψ(Q).

,>k

where h_k(q_k) is the “generalised Hamiltonian” of the k-th component in the absence of
interaction with the degrees of freedom q_k, V_k, (q_k,q_,) is the “interaction potential”
between the k-th and ,-th components, Ψ (Q) is the system state-function depending on
all degrees of freedom, Q≡ {q₀,q₁,...,q_N}, E is the generalised Hamiltonian eigenvalue,
and summations are performed over all (N) system components. The “Hamiltonian”
equation form does not involve any real limitation and can be rigorously derived, in a
self-consistent way, as a universal expression of real system dynamics [9,11,12], where
generalised Hamiltonians express suitable measures of complexity defined below. One

can present eq. (1) in another form, where one of the degrees of freedom, for example
q₀ ≡ ξ, is separated because it represents an extended, common system component or

where from now on Q≡ {q₁,..., q_N} and k,,≥ 1.

The most suitable problem expression is obtained in terms of eigenfunctions
{φ_kn_k (qk)} and eigenvalues {εn_k} of non-interacting components:

hk ⁽qk ^)o. ⁽qk ) = ^εn_k Ψknk ⁽qk ).

^ψ (^ξ. Q )=   ∑   ψn (qо > n₁ (qι) ψ2 n ₂ (q2)--VNn_N (qN )≡∑ψn(ξ)Φn(Q), (4)

n ≡( nɪ, n 2,..., nN )                                                                         n

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