Page 134 - 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 a large amount of bacteria density One interesting scenario is the stationary one with
might be described by micro-forces so that a ( , ) = . In this manner one can assume the

combined description based in Newton forces and following: ( ) / = − in the second term in
electrodynamics can add relevant information to the right side of Eq. 3, yielding:
the process of substrate degradation. In cases of
complex dynamics, of course the usage of diffusion ( , ) = − ⟹ ( , ) = − ∫ . ( )
equation can be obsolete so that an upgraded ( ) ( )
version would have to be needed. For example, one In this manner the Keller-Segel equation with
can see the work of Rosen in 1984 [6] characterized ( ) = one gets:

by having differential equations inspired by the

Navier-Stokes scenario in order to propose a theory [ ( , ) ] + [ ( , ) ] − ( ) [ ( , ) ] = . ( )

of bacteria chemotaxis. The contribution of this
paper is in essence the reformulation of the Keller- 2.1 Essential Assumptions
Segel equation containing the integer-order Bessel
functions. With this an analysis about the Eq.5 lends itself to being operated towards two
implications of electrodynamics is derived. The rest known differential equations. Although one can
apply unphysical procedures, it is clear that in a first
of this paper is structured as follows: In the second instance the Bessel structure can be a good
section, the Bessel-Keller-Segel equation is derived. candidate. In this manner, in order to construct an
Once this equation is established then in the third
section the eletrodynamics equations are derived in integer-order Bessel differential equation [9], one
a closed-form manner. Here the instantaneous needs to impose at the rigth-side the following:

electric current is presented. It is seen that the ( )
resulting distributions exhibit a certain similarity ( ) = − ( − ),
with the discharge of a typical RC circuit. Finally, in that implies that there is a direct relation between
fourth section the conclusion of the paper is the function ( ) and the integer numbers . On the
presented. other hand one can take the definition of Eq. 4 to be
inserted in Eq. 6 yielding:
2. EXTENSION OF THE KELLER-SEGEL
EQUATION − ( ) [ ] = ( − ), ( )
( )
Actually, one can exploit Eq. 1 in many ways by
which one can extract the dynamics of any where the signs have been canceled in both sides.
aggregation of bacteria in different scenarios while Subsequently by applying the derivative in a
under interaction with substrates [7], [8]. Bacteria straightforward manner then Eq. 7 can be read as:
as a biological unit can require particular ( ) −
capabilities in order to guarantee an optimal − ( ) [ ] =
colonization. Although it is not clear whether ( )
bacteria dynamics behave as a linear or nonlinear
system, throughtout this document it will be − [ ( ) − ] = (1 − ) ( )
assumed the linear assumption that allows for exact ( )
extensions.
yielding a first-order differential equation that can
Thus, for instance consider the case where ( ) = be written as:
( ) ≡ ( ) and ( ) = ( ) ≡ ( ) acquiring an
explicit dependence on “ ” then from Eq. 1 one ⟹ − ( ) = ( ) ( − ). ( )
arrives at:

( , ) ( , ) ( , ) It was assumed the approximation given by ≈ ,

= ( ) [ ] − ( )[ ]
(resulting in a toy model for substrate dynamics

that appears to be linear with ). It allows a direct
− ( )[ ( , ) ]. ( )
solution of first-order differential equation that can
be written as:

= ( − ) ⟹ ( ) = [ ( − )]. ( )

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