Macroscopic Growth of Bacteria Precisely Follows The Solution of The Diffusion Equation With Boundary Conditions

Volume 6, Issue 10, October – 2021 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
Macroscopic Growth of Bacteria Precisely Follows

The Solution of the Diffusion Equation with
Boundary Conditions
1-
Dr Ismail Abbas, lecturer at MTC, Cairo University.
2-
Sherif Ismael, MSc, Faculty of Medicine, University of Cairo.
3-
Nora Abbas, PhD Prof Ass at the Faculty of Medicine, University of Cairo.
Abstract:- We provide an experimental proof showing The exponential growth, which varies among bacteria,
that the macroscopic exponential growth of bacteria on a is controlled by many environmental conditions such as
delimited 2D planar surface follows precisely the same temperature, humidity, oxygen content, acidity and by the
solution of the heat diffusion equation with the source / nature of the bacterial species itself.
sink term and the prescribed boundary conditions.
Surprising enough, the macroscopic growth of bacteria
The B-chains previously used successfully in solving the on food surface follows the resolution of the mathematical
heat equation can be applied to solve the complicated PDE diffusion equation with subscribed boundary conditions
resulting from the growth of bacteria. which is the subject of this article.
The in-depth study of experimental microbiology II. THEORY AND EXPERIMENTAL RESULTS
and the study of theoretical mathematical physics are
essential to reveal more characteristics of the growth of A. The theoretical vision of the subject
bacteria in the bounded 2D and 3D geometric space. A recent theoretical study [1] proposed that the
spatiotemporal bacterial growth / decay, n (x, t) follows the
I. INTRODUCTION same trajectory as that of the partial differential heat diffusion
equation, i.e. ,
There are many articles explaining the growth and decay
of bacteria on a microscopic level that is inspecting the time ∂n (x, t) ∂t = D∂2n (x, t) ∂x2 + r n (x, t) (1 - n (x, t)) / k . .
variable only but not its propagation in a macroscopic space. . . (1)
To our knowledge, there are only experimental and In normal conventions.

theoretical studies on the growth / decay in the number of In many real-world scenarios, including infections,
bacterial cells n in the time domain n = n (t). bacterial populations spread through 2D and 3D
configuration space. This process could be modeled using the
this is probably the first rigorous test aimed at finding a Fisher-Kolmogorov equation. [1]
relationship between the macroscopic experimental bacterial
spatiotemporal growth observed in microbiology laboratories However, there is no analytical solution for the general
n (x, t) and the theoretical spatiotemporal mathematical PDE case of the diffusion equation 1 or numerical solution in the
in mathematical physics, i.e. say combining microbiology 2D/3D space of Nabla ^ 2 especially when the boundary
and physical mathematics in a rigorous collective study. conditions 2D / 3D and the source / sink term interact.
Bacterial microbes are single-celled microorganisms Concerning the theoretical aspect which is the
lacking a nuclear membrane, metabolically active and mathematical solution of equation 1, we propose to apply the
dividing by binary fission. they are a major source of disease, so-called matrix chains B successfully used in the heat
medicine and food. diffusion equation with fixed Dirichlet boundary conditions
[2] and also with dissipative conditions at the limits of free
Bacterial growth is the proliferation of bacteria into two absorption [3].
daughter cells, in a process called binary fission. Provided
that no event occurs, the resulting daughter cells are B. The experimental vision of the subject
genetically identical to the original cell. Therefore, We performed an experiment with the macroscopic
exponential bacterial growth occurs. growth of bacteria on flat surfaces of high quality Egyptian
bread maintained at 4 centigrade, pH of 7 with NP air and
Bacterial colonies progress through four main phases of humidity.
growth: the lag phase, the log or exponential phase, the
stationary phase and finally the death or decay phase.
IJISRT21OCT596 www.ijisrt.com 1068

ISSN No:-2456-2165
We found that the macroscopic growth of bacteria equation 1, Fig. 1 with the actual square or circular BC
follows the mathematical solution of the PD diffusion prescribed.
Fig. 1:- Mathematical solution for the 2D diffusion equation with circular boundary conditions.
Below is the photo showing the experimental results of the case for two concentric circles of bacteria 2.5 and 10 centimeters
in diameter.
Fig. 2:- Macroscopic growth of bacteria.

ISSN No:-2456-2165
7 days Growth of bacteria on a flat food surface (high quality Egyptian bread) maintained at 4 degrees Celsius with a pH of 7,
normal air and humidity.
The analogy between Fig. 2 and Fig. 1 is obvious.
Fig. 3:- The macroscopic death phase of the bacterial experiment of Fig. 2.

ISSN No:-2456-2165
Three weeks decay of bacteria on a flat food surface
(high quality Egyptian bread) maintained at 4 degrees Celsius
with a pH of 7, normal air and humidity
Note that macroscopic shrinkage and drying in rigor

mortis propagates in the opposite direction to the growth
phase.
III. CONCLUSION
The current article is a small step on a long road.
It shows that the macroscopic exponential growth of

bacteria follows the spatiotemporal partial differential
diffusion equation with boundary conditions.
We propose that a more in-depth study of experimental

microbiology and a theoretical study of mathematical physics
is important to reveal more features of bacteria growth in 2D
and 3D geometric space with boundary conditions.
REFERENCES
[1]. Published online 2018 Oct 1.

doi: 10.1088/1361-6633/aae546
[2]. I. Abbas , IJISRT, A Numerical Statistical Solution to
Laplace and Poisson Partial Differential Equations,
Volume 5, Number11, November - 2020
[3]. I. Abbas , IJISRT, A Time-Dependent Numerical
Statistical Solution of the Partial Differential Heat
Diffusion Equation, Volume 6, Issue 1, January - 2021.
[4]. I. Abbas ,Theory and design of audio rooms-
Reformulation of Sabine Formula, IJISRT review, Vol
6 Oct 2021 , Researchgate, October 2021

Macroscopic Growth of Bacteria Precisely Follows The Solution of The Diffusion Equation With Boundary Conditions

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Volume 6, Issue 10, October – 2021 International Journal of Innovative Science and Research Technology

Macroscopic Growth of Bacteria Precisely Follows

To our knowledge, there are only experimental and In normal conventions.

IJISRT21OCT596 www.ijisrt.com 1068

Fig. 2:- Macroscopic growth of bacteria.

IJISRT21OCT596 www.ijisrt.com 1069

The analogy between Fig. 2 and Fig. 1 is obvious.

IJISRT21OCT596 www.ijisrt.com 1070

Note that macroscopic shrinkage and drying in rigor

The current article is a small step on a long road.

It shows that the macroscopic exponential growth of

We propose that a more in-depth study of experimental

[1]. Published online 2018 Oct 1.

IJISRT21OCT596 www.ijisrt.com 1071

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