Page 135 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 3 – Internet of Bio-Nano Things for health applications

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 3

In this manner the chemotactic coefficient is bacteria motility properties such as cooperative
expressed in terms of “ ”. A direct solution for it behavior. It is noteworthy that the negative values
yields the exponential form and depends directly on of Bessel functions are not describing physical
the substrate density “ ”. Therefore, one arrives at: solutions. Because of this, the positive values of
bacteria density might be interpreted in terms of

⟹ ( ) = [ ( ) ( − )] ( ) cooperative and competitive population, depending

on the width of distribution. Clearly this requires
the option of square of the Bessel functions that
provides only physical meaning. Therefore from Eq.
13 one arrives a the solutions given by:

( , ) ∝ | ( , )| 2 ( )

Fig. 1 – The normalized chemotactic coefficient from Eq. 11 for
up to 4 values of number “ ”, denoting the order of Bessel
functions. For this it is assumed the unidimensional

representation of s(x) = [ . ] [ . ∗ ].
2.2 Charged Bacteria Density Fig. 2 – The normalized bacteria density ( , ) as a function of
distance (in arbitrary units) for the first 3 orders of Bessel
In Fig. 1 up to 4 different distributions of the functions and their tentative interpretation in terms of
normalized chemotactic coefficient are displayed cooperative population.
under the assumption that the substrate density has In Fig. 2 up to three possible bacteria density are
the quasi-stochastic form given displayed. For this a normalization function was
by ( . ) ( . ) [10]. This opted, and it reads 2.9 √ − × where is

mathematical approach comes from the fact that the
existence of periodical manifestation of electric about the 5% of for this exercise (this is assumed

behavior would exhibit both attraction and both free parameters and , to illustrate Fig. 2).

repulsion, depending the state of pair: bacteria- Clealrly, both can be changed in more analytic
substrate. scenarios. Therefore the density spectra for s fixed
time “T” reads:
Thus, positive and negative values with rapid 2
oscillation along the distances are expected to be ( , ) = √ − × | ( − × , , )| ( )
allowed by the motility of bacteria. One can see that In fact, inspired at the criterion of [3] in which the
the higher order exhibits a well-shaped peak. bacteria densities have a steady-state profile
centered in − × being this dependent on the
2.3 The Bessel-Keller-Segel (BKS) Equation
integer “ ”. One can see that while the order of
Therefore from Eq. 5 to Eq. 9 one arrives at a novel Bessel function increases then one would expect the
version of the Keller-Segel equation called BKS cooperative population moves with distance; as
(Bessel-Keller-Segel), that is written below as: seen in Fig. 2 the distributions moves to the right.

( , ) ( , ) A compelling argumentation of why one calls

[ ] + [ ] + ( − ) ( , ) = ( )
cooperative population to the distribution (yellow
that is solvable for the bacteria density yielding color), can be directly seen in the width of each curve.
imminently the integer-order Bessel functions: Thus, when it is larger than the others, one arrives to
the fact that bacteria join each other to perform a
( , ) ( )
( , ) = ( , ) = subsequent action ("Many make force”). A short
Although a solid interpretation of Eq. 13 might not width is perceived as The number of individuals
be adjusted to the scope of this paper, it is possible along the time of interaction can be obtained from a
to some extent to adjudicate a meaning in terms of simple integration ( ) = ∫ ( , ). Thus one can

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