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APPLICATION OF MULTIDIMENSIONAL TIME MODEL TO DYNAMICAL RELATION OF POISSON SPIKE TRAINS IN NEURAL ION CURRENT MODELS AND FORMATION OF NON-CANONICAL BASES, ISLANDS, AND G-QUADRUPLEXES IN DNA, MRNA, AND RNA AT OR NEAR THE TRANSCRIPTIONMichael Fundator
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Article Type: Research Article
Article Citation: Michael Fundator. (2021). APPLICATION OF MULTIDIMENSIONAL TIME
MODEL TO DYNAMICAL RELATION OF POISSON SPIKE TRAINS IN NEURAL ION CURRENT
MODELS AND FORMATION OF NON-CANONICAL BASES, ISLANDS, AND G-QUADRUPLEXES IN
DNA, MRNA, AND RNA AT OR NEAR THE TRANSCRIPTION. International Journal of
Research -GRANTHAALAYAH, 9(1), 7-15. https://doi.org/10.29121/granthaalayah.v9.i1.2021.2918
Received Date: 27 November 2020
Accepted Date: 21 December 2021
Keywords:
Kolmogorov-Chentsov Continuity Theorem
Fokker-Plank
Stochastic Differential Equation
Translation and
Transcription
Neural Networks
ABSTRACT
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:
p =
p(v, g; x, t)
(1)
to find a
neuron at time t within the xth the
coarse-grained (CG) patch that the
cortex is tiled with and are sufficiently small for the cortical architecture
not to change systematically across it, but are sufficiently large to contain
many (hundreds) of neurons) with voltage (v, v+ dv) and conductance (g, g + dg) that in quantum theory could be
considered probability current, involves writing of the FP SDE for the
probability evolution process and flux rate equation:
J(v,g;t)+
(2)
Where 
is the “leakage” time
constant, 
is the time constant
of the AMPA synapse (mediating transmission of electrical or chemical signal in Nervous System) , f is the synaptic strength from LGN connections, N is
number of neurons, and S is the cortico-cortical coupling strength.
J(v,g;t)
= 
(3)
is the flux along ν, v 0(t) is the (temporally modulated) rate for the Poisson spike
train from the LGN
= 
(4)
Is the
population-averaged firing rate per neuron in the CG patch, determined by
the J-flux at the threshold to firing with normalized,
dimensionless potentials related to original Hodkin-Huxley
model V T =
1, V R = 0, and V E = constant.
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 d;
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 (Wt
= (
linear multidimensional FP SDE is
derived below (12-17) and has a form [4-17]:
= [ -
(
) + 
(
) ] 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.
Figure1 (a and b): Possible appearances of transient Poisson
spike train.
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
Theorem1. BM is
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.
Proposition2. The continuous modification of
the solution of FP SDE is a function of BM.
Proposition3. The continuous modification of
the solution of FP SDE is transient if BM is the multidimensional BM.
Definition4. Multidimensional stochastic
process (Wt = ( is
called multidimensional
BM if the processes (
are independent BMs.
4.
RULES
FOR FINDING CONTINUOUS SOLUTIONS TO FOKKER-PLANK SDE [36-40].
Rule1. If there is a transient condition
the solution should be sought in form of function of multidimensional BM.
Rule2. Jump processes are associated
with Poisson processes [25]. If Poisson spike train is supposed to be
terminated, then Rule1 should be applied. Otherwise, time interval could be
divided into smaller intervals that would lead to m-dimensional time
parameter t = (t1…
tm) and modification
of Kolmogorov’s continuity criterion [23 p3]:
Ck
|t-s
for all 0 T;
(9)
such
that si, ti k,
i = 1,…,m.
Rule3. Many solutions of Hodkin- Huxley model look like
periodical with period T. Then the solution is sought in form of product of
sums of trigonometric functions with period T and Brownian bridge (BB) over
time interval [0, T] from x to y. BB is a Gaussian process that can be described
as [23]:
Bx,T,y 
{x + (y-x)
+ Wt - Wt ; 0 
T}.
(10); Or
dBt = 
dt + dWt ; 0 T,
and B0 = x
(11).
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.
Rule4. For special problems
n-dimensional Bessel process could be used [23]:
(12)
Or some
functions of this process [23].
Rule5. For different techniques of
finding appropriate number of dimensions and subsequent reduction see [6, 7,
and 9].
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 (Wt. This in its turn implies
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 G4C2 repeats in contrast to a normal person that usually has below 8 G4C2s. As in the 1.
· G4s tend to occur near end 3’.
· Hypothesis 4. Ions that are found inside G4 structure imply ion currents which are hardly detectable, but depend as in Hypothesis 1 on different factors including the memory and the information transmitted by transient spike trains and was stored in the inside or surrounding human body systems and later contribute to the process of transcription and translation.
8. APPLICATIONS OF MAIN PROPOSITIONS 1, 2, AND 3 HYPOTHESES 1, 2, 3 AND 4
Theoretical 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. CONCLUSION
Translation 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 FUNDING
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
CONFLICT OF INTEREST
The author have declared that no competing interests exist.
ACKNOWLEDGMENT
The author is grateful to
Yehuda Goldgur from Sloan Kettering Cancer Center for the initial review and very
valuable suggestions about this article.
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