Kolmogorov-Chentsov Continuity Theorem Fokker-Plank
Stochastic Differential Equation Translation and
Transcription Neural Networks Ground breaking application of mathematics and biochemistry to explain formation of non-canonical bases, islands, G-quadruplex structures, and analog bases in DNA and mRNA at or near the transcription with connection to neural networks is implemented using statistical and stochastic methods apparatus with the addition of quantum principles. As a result the usual transience of Poisson spike trains (PST) becomes very instrumental tool for finding periodical type of solutions to Fokker-Plank (FP) stochastic differential equation (SDE). The present study develops new multidimensional methods of finding solutions to SDE. This is based on more rigorous approach to mathematical apparatus through Kolmogorov-Chentsov continuity theorem (KCCT) that allows the stochastic processes with jumps under certain conditions to have γ-Holder continuous modification, which is used as basis for finding analogous parallels in dynamics of formation of CpG and non-CpG islands (CpGI or non CpGI), repeats of G-quadruplexes, and non canonical bases during DNA (de)- methylation and neural networks.
## 1. INTRODUCTION“Thousands
of diseases are produced by genetic defects in channels, including many
diseases of profound importance, like cystic fibrosis, epilepsy, atrial and
ventricular fibrillation, and so on, as documented in many papers …. among
thousands of others. Many of these diseases are caused by problems in the
construction of channels, or the insertion of channels in the wrong places in the
wrong cells, or in the regulation and control of channels”. (Bob Eisenberg “Crowded Charges in Ion Channels”.) Recent
numerous applications of FP SDE for flux rate to the to describe the
mathematics of neural networks were introduced more than 60 years ago [21]. FitzHugh’s
consideration of stochasticity in neural networks was based on earlier works
including Hodkin’s article after Kramers
introduced FP SDE to the theory of rate of chemical reactions. He used notions
of equilibrium, energy barrier, and memory friction. Focus on development of
kinetic and dynamical theory for integrate and fire (I&F) neurons with
application to simple (with inhibitory usually linear responses) and complex
(characterized by strong cortical nonlinear and significant second harmonic
responses excitation) cells (determined by balance between cortico-cortical
input and lateral geniculate nucleus (LGN)) in visual cortex provided
qualitative intuition for dynamic phenomena related to transitions to bistability and hysteresis, where system jumps back and forth between the two branches of
stable states for different, critical values of some control parameter that can
result in optical binding. ## 2.
FOKKER-PLANK
STOCHASTIC DIFFERENTIAL EQUATIONS FOR FLUX RATE
Large
part of neural networks analysis consists of a study of reduction of infinite
models to finite models that include different types of bifurcations.
Bifurcations occurring in neural networks include double period bifurcations, Hopf bifurcations, saddle-node bifurcations, torus
bifurcations, and homoclinic bifurcations. Thus, the analysis is related to
numerous quantum theoretical assumptions. For example in the visual cortex the kinetic equations for
evolution of the probability density:
to find a
neuron at time t within the x
is the flux along ν,
Is the
population-averaged firing rate per neuron in the CG patch, determined by
the The
evolution processes are certainly the stochastic processes with jumps[28], e.g. transitions between metastates
and to bistability and hysteresis, Poisson spike
train etc…, and only application of Kolmogorov-Chentsov continuity theorem [20, 21, 23, 24] assures the
existence of sample continuous modification of
stochastic processes with the same probability law: () = 1;
(5) under the
conditions of existence of the constants , C for some metrics
C
|t-s for all 0 T;
(6) and
the modified process would be - Holder continuous for every 0 < < . The importance of Kolmogorov’s
continuity criterion for the whole theory of Brownian Motion Stochastic Calculus including
stochastic differentiation and integration and Stochastic Differential
Equations is emphasized in [20], where the authors used this criterion for the
constriction of Brownian motion. The
presence of equilibrium and voltage threshold of spike activation justifies the
existence of smooth modification for the solution of Equation (2)
in the presence of Poisson spike train from the LGN for the majority
of experimental conditions. For
n-dimensional process based
on multidimensional BM ( = [ -( ) + ( ) ] p(,t) (7) Operators ( ) and ( ) (8) are called drift vector and diffusion (correlation
or dispersion) matrix respectively and include memory friction or viscosity coefficient that
is involved in derivation of FP SDE. The Poisson spike train that was mentioned in
the relation to initial conditions in Equations (3) and (4) is transient in
most applications of the Hodgkin-Huxley model that makes it highly instrumental
tool for finding solutions to different SDE, as in the (Figure1(a and b)
) a. b.
This could be explained
by the argument of [22]. Simulation of the Hodgkin-Huxley model depending on the different steps
increase in current produces outcomes such as: 1)
Damped
oscillation, 2)
A
single spike that is followed by a transient spike, and 3) Sustained firing. Consideration
of the case (2) can be applied to find solutions to many other examples of SDE
with or without jumps. Because the spike is transient the following theorem for
the multidimensional Brownian motion (BM) can be applied [20, 24] ## 3.
TRANSIENCE
Point
recurrent in dimension d=1. For reference and discussion look [19]. Neighborhood
recurrent, but not point recurrent in d=2. Transient
in d 3.
## 4.
RULES
FOR FINDING CONTINUOUS SOLUTIONS TO FOKKER-PLANK SDE [36-40].
C such
that s
B dB It
can be observed that this process is a solution to the above SDE
and another remarkable property of BB is that it can be well approximated by
simple combinations of multidimensional
BMs.
(12)
Or some
functions of this process [23].
In the above example (1)-(4) the application of FP SDE to neural networks implements reduction of dimensions to (1+1)D
from originally used 3D or (2+1)D
in neural networks [17] due to
involvement of Poisson spike trains. ## 5. MAIN PROPOSITIONS 1, 2, AND 3
Main Proposition 1
According to Theorem1, existence of additional dimensions in the
continuous modification of solution of FP SDE which according to Proposition2
is a function of BM. Main Proposition 2 The same mathematical reasoning
in construction of Fokker-Plank equation for the rate of transcription and
translation in different fragments of DNA, messenger RNA (mRNA), and RNA and
multiplicity of other experimental and theoretical factors suggest existence of
the same additional dimensions of Main Proposition 1 of the continuous
modification of solution of Fokker-Plank equation in neural network model. Main Proposition 3 These additional dimensions that
are mentioned in Main Proposition 1 and Main Proposition 2 are the factors in the formation of islands (CpGI and non-CpGI) of different
analog bases, e.g. CpG and non-CpG (CpA, CpC, and CpT)
sites that are found in bigger proportions at or near the transcription ends
defying statistically supposed distributional assumptions, and non-canonical
base pairs contributing to disruption of double helix in DNA. Most, perhaps all CpGIs
are usually related to the sites of transcription initiation, including
thousands that are remote from currently annotated promoters [30]. ## 6.
DNA TRANSLATION AND TRANSCRIPTION AND HYPOTHESES 1 AND 2 [26,
36-40].
The addition of methyl groups to DNA is called
DNA methylation, whereas deletion is called demethylation. Both processes are
studied in cancer research and other genetic related diseases. To understand the necessity of application of
quantum type principles to proteins and further facilitate the discussion of
what kind of principles should be applied, it would be appropriate to mention
the following questions and problems in investigation of proteins, DNA, and RNA
structures [26]. In contrast to RNA, DNA has a double helix structure
under Watson-Crick base pairs model that identifies standard base pairs (e.g. A-T, G-C). Non-canonical base pairs contribute to
disruption of double helix in DNA. In the beginning of investigation
it was supposed that the approach to structure or sequence of bases or other
properties should be probabilistic or statistical because all the alternatives
seemed to be equally possible. This led to some distribution assumptions about
70 years ago which were followed by careful experimental investigations until
different authors became convinced that the sequences of the bases do not
follow any known probability distribution function, even in the fragments of
DNA. This resulted in additional
investigation and served as guide to researchers until there were found two
additional analog bases: 5 methylC (the fifth base
analog) and 5 hydroxymethylC
(the sixth base). However, the belief that the structure would be described by
some Probabilistic or Statistical Mechanics Principals was still very
disorienting. This situation caused some additional questions that were finally
resolved with finding two more analog bases: 5-formylcytosine and 5 carboxylcytosine
which are actually
versions of cytosine that have been modified by Tet proteins, molecular
entities that are thought to play a role in the DNA demethylation and stem cell
reprogramming. Much in
contrast the dynamical approach makes the situation quite different, and
certain quantum principles that were applied by Kramers
during introduction of FP SDE to the theory of rate of chemical reactions
suggest similar application of FP SDE to the rate of translation and
transcription of different fragments of DNA and RNA. This in its turn would
suggest that the formation of islands and informal pairs of bases, analog
bases, or G-quadruplexes is a result of some multidimensional stochastic
process involving a continuous modification of solutions of FP SDE to the rate
of translation and transcription. Similarity in application of FP SDE to the
rate of translation and transcription in DNA and to neural ions model with many
other different facts from these theories implies that formation of islands and
informal pairs of bases in DNA depends on spike trains and the like phenomena: 1) The process of transcription involves translation from certain parts of DNA that is double stranded to mRNA, where the fragments are modified and are bind to a small RNA. As a result, mRNAs bind metabolites with untranslated regions (UTR) of RNA, and then the modified parts are copied back to DNA. These parts and stages of translation and transcription and translation naturally serve as memory friction or viscosity in the formation of Fokker-Plank equation. There are also additional steps that make the process sufficiently complicated to consider application of stochastic analysis. However, as mentioned previously such considerations are insufficient, but 2) There are certain constraints in the transcription process that make it highly similar to the corresponding subject of the velocity of chemical reactions in the sense that the translation has to go through some kind of a barrier, as in the case of the particle that has to go over potential barrier. To make a partial list: ·
The
steps accounting for primary transcription involve consist from
multiple molecular interactions followed by the transcription of DNA. ·
Depending
on cellular activity transcripts from DNA contribute to different RNA
translations into functional proteins. ·
Sequences of DNA in their turn control initiation of
RNA primary
transcription. The a.
and b. constraints correspond to probability of escape barrier for the particle
for reaction activation in the theory of the rate of chemical reactions, c.
corresponds to the notion of equilibrium, besides that these constrains also
contribute to the corresponding notion of memory friction that is included in operators ( ) and ( )
(13) For
n-dimensional process based
on multidimensional BM linear FP SDE as in above (7) [4-17] = [ -( ) + ( ) ] p(,t) (14) For
example, in 1D velocity of Brownian particle =
(x,v) moving in U(x)
potential is given by Langevin equation:
(15) Where x,
v, and m are the position, velocity, and a mass of the particle and is noise, and is a coefficient of viscosity or memory
friction. The Langevin equation after
substitution into probability conservation equation would be written in form of
FP SDE: =
(16) Hypothesis
1 that can be viewed as explanation of Main Propositions 1, 2, and 3. The
information transmitted by transient spike train is stored in the inside or
surrounding human body systems and contributes to the memory friction
coefficient in the process of transcription and translation, and the rate of
transcription and translation can be described by multidimensional Fokker-Plank
equation, based on multidimensional BM (W Hypothesis
2 The number of dimensions that are involved in formation of Fokker-Plank
equations in ion channels model and in the processes of transcription and
translation of DNA and RNA can be considered equal. ## 7. FACTS AND EXAMPLES SUPPORTING MAIN PROPOSITIONS1, 2, 3 AND HYPOTHESES 3 AND 4.Most, perhaps all CGIs are usually related to
the sites of transcription initiation, including thousands that are remote from
currently annotated promoters [29]. This fact in combination with Main Propositions 1, 2, and 3,
Notion 1 and Hypothesis 1 imply Hypothesis 3 The intensity of
distribution of CpGIs around transcription initiation
can be distributed according to some probability distribution, and, if found to
follow some stochastic law, could help in further investigations. DNA
methylation in many instances is transient [29,32,33]. Recent
studies suggest high impact of DNA methylation on memory [35]. DNA
methylation depends on stress [29]. Contrary to the expected
that the connectivity of organs would require biochemical similarity (see
adipose tissue [32] in #4 in Applications of Main Propositions 1, 2, and 3), only some of
the total possibility of genes are expressed in any given tissue. For example,
a protein that is active in a nerve cell is not expressed in the
liver. [33] The most
profound examples to support the theory of CpG and GpC
islands and non regular bases formation due to
additional dimensions involvement in transcription and translation processes is
the discrepancy of DNA
methylation levels in twin children or astronauts under changing environmental
conditions in questions if the changes are transient or long-lived. Multidimensional
approach to skeletal muscles and biological motion that contains information
about actions, intentions, and emotions [34], and their effect through exercise
on methylation process. ·
Patients with amyotrophic
lateral sclerosis or frontotemporal dementia develop from 500 to few thousands
G ·
G ·
Hypothesis 4. Ions that are
found inside G ## 8. APPLICATIONS OF MAIN PROPOSITIONS 1, 2, AND 3 HYPOTHESES 1, 2, 3 AND 4Theoretical approach based on multidimensionality of
Brownian motion in formation of Fokker-Plank equations for ion channels model and rate of transcription
and translation of DNA and RNA allows multiple applications: 1) During
first stages of genetic diseases. 2) In
palliative care and opens a new approach of computational study during palliative
care. 3) DNA
methylation can carry effective predictive functions, as in the different types
of diabetes due to complex correlation of DNA methylation and insulin
gene expression. 4) Provides
access to multiple discoveries in the direction of tissue analysis. 5) Analysis
of the above 3 applications suggests that the information from the transient
spikes is stored in electro-magnetical or bioossilations contributing to atrial fibrillation and similar diseases
and acts only after long period of time depending on different factors. For example adipose tissue is essential in regulating whole body
energy metabolism and besides adipocytes, adipose tissue contains connective
tissue matrix, nerve tissue, stromovascular cells, and immune cells, and under
some conditions it contributes to suppression of genes transcription[31]. ## 9. CONCLUSIONTranslation and transcription of DNA and RNA through
transience, hidden dimensional consideration, memory friction, and other
factors can be highly connected to neural networks and this connection can be
mathematically modeled. ## SOURCES OF FUNDINGThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. ## CONFLICT OF INTERESTThe author have declared that no competing interests exist. ## ACKNOWLEDGMENTThe author is grateful to
Yehuda Goldgur from Sloan Kettering Cancer Center for the initial review and very
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