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Relevance of Quantum Mechanics
in Bacterial
Communication
Credit Seminar I
INTRODUCTION
◎ We think that a multiple of bacteria are stronger than a
few and thus by union are able to overcome obstacles
too great for the few.
◎ There are many situations when the bacterial population
behave co-operatively and recognize self and non-self
which can be highly advantageous.
◎ In the contexts of symbiosis, niche adaptation,
production of secondary metabolites and for facilitating
population migration.
◎ These bacterial communities are glued together by
EPS, which is called a biofilm.
Bacterial
communication
Chemical
signaling
Electrical
signaling
CHEMICAL SIGNALING
◎ Autoinducers are wide variety of molecules, which are
used by bacteria for inter- species and intra- species
communication process.
◎ When a threshold concentration of the signaling
molecules is achieved, a coordinated change in
bacterial collective behavior is initiated.
◎ This bacterial collective behavior is known as quorum
sensing by which bacterial cell integrate in order to
determine their optimal survival strategy.
◎ Characteristic features to trigger the networking system-
◉ The production of the quorum sensing signal takes
place during specific stages of growth, under certain
physiological conditions, or in accordance to
environmental changes.
◉ The quorum sensing signal diffuses in the
extracellular space and is recognized by specific
bacterial receptors.
◉ The accumulation of a critical threshold
concentration of the quorum sensing signal initiates
a firm response.
◉ The cellular response further offer beyond
physiological changes that metabolize or detoxify
the molecule.
QUORUM SENSING
◎ Quorum sensing coordinates the gene expression,
when the bacterial cell population has reached a high
cell density.
◎ Quorum sensing is responsible for mediating a variety
of social activities in biofilms, which include the
swarming motility, biofilm dispersion, biofilm growth and
antimicrobial resistance etc.
◎ It is found that quorum sensing regulating EPS
produced during biofilm formation.
◎ Bacterial coordinated phenotypes are driven via quorum
sensing systems, beneficial only at certain cell
densities.
◎ Quorum sensing can function as a way to outcompete
neighbors in patches occupied by many different
genotypes.
◎ In these communities, competing microbial genotypes
gradually segregated over time leading to positive
correlation between density and genetic similarity
between neighboring cells.
◎ Growing aggressively made quorum sensing genotypes
a match for competitors.
◎ During growth, cells secrete autoinducers, that
accumulate in the environment and high autoinducer
concentration around cells induces expression as well
as promotes metabolical traits.
◎ Cells can tune the expression of density-dependent
phenotypes, like virulence factors or secreted enzymes.
Quorum sensing
Gram negative
Acyl homoserine
lactone (AHL) and
quinolone
Gram positive
autoinducer
peptide (AIP),
ranges 5-34 amino
acids in length
ELECTRICAL SIGNALING
◎ The electrical communication within bacterial
communities occurs through spatially propagating
waves of potassium.
◎ When these communities grow larger, the supply of
nutrient to the interior cells becomes limited because
the nutrient consumption is increasing.
◎ At the same time, this nutrient consumption is
associated with the growth of multiple layers of cells in
the biofilm periphery.
◎ The conflict between
starvation and protection
is resolved through
emergence of long range
metabolic co-
dependence between
interior and peripheral
cells and gives rise to
collective oscillations
within biofilms.
◎ Synchronized collective
oscillations of membrane
potential and the
bacterial ion channels
activate the electrical
communication.
◎ In 2017, it has been discovered that two B. subtilis
biofilm communities undergoing metabolic oscillations
become coupled through electrical signaling and
synchronize their growth dynamics.
◎ It is confirmed that biofilms resolve this conflict by
switching from in phase to anti- phase. Different biofilm
communities take turns consuming nutrients.
◎ Thus distant biofilms can coordinate their behavior to
resolve nutrient competition through time-sharing.
SYNCHRONIZED
GENETIC OSCILLATORS
◎ In biology, a vast range of intercellular coupling mechanisms lead to
synchronized oscillators which govern fundamental physiological
processes such as cardiac function, respiration, insulin secretion, and
circadian rhythms.
◎ Synthetic biology can be broadly parsed into efforts aimed at the large-
scale synthesis of DNA and the forward engineering of genetic circuits
from known biological components
◎ In the area of DNA synthesis, pathways have been perturbed and
replaced in an effort to understand the network motifs and transcriptional
regulatory mechanisms that control cellular processes and elicit phenotypic
responses
◎ Here, the original toggle switch and oscillator have inspired the design and
construction of circuits capable of controlling cellular population growth,
generating specific patterns, triggering biofilm development, shaping
intracellular noise, detecting edges in an image, and counting discrete
cellular events.
◎ A unifying theme for most of the genetic circuit studies is a
particular focus on dynamical behavior. Thus the circuits are
constructed and monitored in single cells, typically with fluorescent
reporters, and new measurement technologies are often developed
in parallel.
◎ Tools from the fields of nonlinear dynamics and statistical physics
are extremely useful in both the generation of design specifications
and for careful comparison between experiment and computational
model.
◎ The synchronized oscillator design is based on elements of the
quorum sensing machineries in Vibrio fisheri and Bacillus
Thurigensis. The luxI (from V. fischeri), aiiA (from B. Thurigensis)
and yemGFP genes are placed under the control of three identical
copies of the luxI promoter.
CHEMICAL METHODS TO
INTERROGATE
BACTERIAL QUORUM
SENSING PATHWAYS
◎ QS challenges the traditional notion of
bacteria as autonomous agents by
permitting them to function as
multicellular groups and thrive in
specific environmental niches.
◎ Many of the phenotypes can have
significant impacts on human health,
agricultural yields, industrial
manufacturing, and ecology.
◎ Binding of the AIs to their target
receptors activates the transcription of
genes required for QS phenotypes,
along with those associated with AI
biosynthesis.
◎ Increased production of the AI signal
once a quorum is reached enhances the
sensitivity of the signaling process and
facilitates population-wide
synchronization of the QS-regulated
phenotype.
Combinatorial
approaches
• AHL analogues on
polystyrene resins
• AHL analogues
using small molecule
microarray
• DKP using small
molecule microarray
• AIP analogues on
polystyrene resins
• PQS analogues
Quorum
Quenching
Antibodies
• Designing antibodies
to sequester AIs
• Catalytic antibodies
to inhibit QS
Abiotic polymers
for QS
modulation
• Polymer “Sinks” for
AHL
• Polymer “pools” for
the controlled
release of AIs
Electrochemical
techniques to
study QS
pyocyanin an
electrochemical probe
PQS as an
electrochemical probe
THEORY
◎ The densely packed bacterial populations develop a
coordinated motion on the scales length 10µm to
100µm in comparison to the size of each single
bacterium of order 3µm when the bacterial cell density
reaches a sufficiently high value.
◎ The finite size of the bacteria indicates the existence of
an intermediate length scale, which leads us to
introduce a source of fluctuation, which is quite different
than thermodynamic fluctuation.
◎ The swimming induced stresses on the bacteria that
can change the local arrangement of bacteria induce
stress fluctuations.
◎ Two different type of noise are present in the bacterial
communication system and dominance of one over the
other depends on the force
◎ F=f/ρg
◉ Where,
o ρ = density
○ f = the volume of the forcing
○ g = the acceleration due to gravity
VISCOSITY AND NON-LOCAL THEORY
◎ The rearrangement of the configuration of the coarse grained
systems produce a noise that gives rise to kinematic
viscosity.
◎ Noisy burger equation:-
where,
u= field
v= kinematic viscosity
ɳ= noise
◎ The Noisy burger equation gives the view of an internal
structure of the complex biological communication
system and viscosity is the property which makes the
bacterial cells stick together into clusters predicted by
Zeldovich approximation, just mimicking gravitational
effect on the smaller scales.
◎ This approximation can describe the general structure
of this nonlinear biological phenomenon. It is to be
mentioned that the origin of viscosity is associated with
the weakly non-local effects in the internal structure of
the system.
◎ The kinematic viscosity plays a vital role in forming the
metastable states of the bacteria responsible for
quorum sensing.
◎ Kwak Transformation is a reaction- diffusion system,
which gives the mathematical framework for the pattern
formation.
Where,
PATTERN FORMATION
AND VISCOSITY
◎ In this multicellular system bacterial cells form different
patterns based on chemical gradients of QSM signal
that is synthesized by quorum sensing bacterial cells.
◎ Furthermore, the mathematical approach is able to
predict how the system behaves if we change the initial
value.
◎ It was observed that the quorum takes place in a certain
range of kinematic viscosity([0.01, 0.32]m /s), which is
considered as very small viscosity of the fluid.
◎ The behaviour changes with the initial data and system
forms different wave patterns.
2
ELECTRICAL
COMMUNICATION AND
NON-LINEAR
SCHRÖDINGER EQUATION
◎ The potassium ions follow non-linear Schrödinger equation.
Where,
ᴪ= wave function of Potassium
ion.
i = imaginary unit
ħ= reduced Planck constant
(1.054*10-34 J.s)
r= position vector
t= time
Ĥ= Hamiltonian operator
◎ This non-linear Schrödinger equation is valid at the level of
ion channel where as the perturbation becomes predominant
at the cellular level.
◎ This fluctuation gives rise to the perturbation on non-linear
Schrödinger equation and we get generalized Complex Ginzburg-
Landau(GL) equation.
Where,
F= free energy
Fn= free energy in normal phase
α & β= phenomenological
parameters
m= effective mass
e= charge of an electron
A= magnetic vector potential
B= magnetic field
µ0= refractive index
◎ This Complex GL equation is used for the description of cellular
communication through the chemical molecules and also needed
to understand the generation of various patterns in Biofilms.
COMMUNICATION
BETWEEN ARTIFICIAL
AND NATURAL CELLS
AND TURING TEST
◎ One can find the solution may narrate biochemical
systems on a continuum where the binary categorization
of alive not alive is replaced by states that are
increasingly lifelike.
◎ Following this path, each iteration of constructing an
artificial cell could be objectively and quantifiably
evaluated in terms of likeness to a target natural cell.
◎ Recent experiment has attempted to reconstruct the well
characterized quorum sensing pathways to build an
artficial cell which can mimic the ability of the natural
cells of V. fisheri, E. coli, P. aeruginosa.
◎ Turing (1950), in his seminal paper pointed out that the
ability of a machine to deceive a judge through textual
communication into believing that the machine is a
person was used to circumvent the problem of defining
intelligence.
◎ A cellular Turing test is possible because all cells can
communicate from quorum sensing pathways in
bacteria to pheromone responses in higher organism.
◎ Moreover these artificial cells containing DNA supports
the transcription and translation prosesses which can
express genes.
◎ The genetic constructs in water-in –oil emulsion
droplets are able to either sense and send quorum
sensing molecules.
◎ Since quorum sensing is directly linked to gene
expression, the next gereration sequencing technology
can be used to quantifiably evaluate the extent of
mimicry.
◎ A futher investigation is required for a much deeper
understanding of life and the cellular Turing test can be
a helful guide to achive this goal
CONCLUSION
◎ Bacteria use chemical signaling molecules, which are called
as quorum sensing molecules (QSMs) or autoinducers.
◎ The ion channels in bacteria conduct a long-range electrical
signaling within biofilm communities through propagated
waves of potassium ions and biofilms attracts other bacterial
species too.
◎ The bacterial communication mechanism is explained by
complex Ginzburg- Landau equation and the formation of
patterns depends on kinematic viscosity associated with
internal noise.
◎ The potassium wave propagation is described by the non-
linear Schrödinger equation.
◎ By adding perturbation to nonlinear Schrödinger
equation one arrives at Complex Ginzburg-Landau
equation which is used to understand the bacterial
communication as well as pattern formation in Biofilms
for certain range of kinematic viscosity.
◎ Here, the perturbation is due to the existence of non-
thermal fluctuations associated to the finite size of the
bacteria.
◎ It sheds new light on the relevance of quantum
formalism in understanding the cell-to-cell
communication.
REFERNCES
◎ Majumdar S, Roy S. Relevance of quantum mechanics in bacterial communication.
NeuroQuantology 2018 Jan 29;16(3).
◎ Majumdar S, Pal S. Bacterial intelligence: imitation games, time-sharing, and long-
range quantum coherence. Journal of Cell Communication and Signaling.
2017;11(3):281-84.
◎ Majumdar S, Roy S, Llinas R. Bacterial Conversations and Pattern Formation.
bioRxiv 2017:098053.
◎ Majumdar S, Mondal S. Conversation game: talking bacteria. Journal of Cell
Communication and Signaling 2016;10(4):331-35.
◎ Roy S, Llinas R. Non-local hydrodynamics of swimming bacteria and self-activated
process. InBIOMAT 2015: International Symposium on Mathematical and
Computational Biology 2016:153-65.
◎ Roy S, Llinás R. Relevance of quantum mechanics on some aspects of ion channel
function. Comptes Rendus Biologies 2009;332(6):517-22.
◎ Ván P, Fülöp T. Weakly non-local fluid mechanics: the Schrödinger equation.
InProceedings of the Royal Society of London A: Mathematical, Physical and
Engineering Sciences 2006; 462 (2066):541-57.
◎ Ribeiro AL, de Faria JP. Weakly nonlocal hydrodynamics and the origin of viscosity
in the adhesion model. Physical Review D 2005;71(6):067302.
◎ Cruz-Pacheco G, Levermore CD, Luce BP. Complex Ginzburg– Landau equations
as perturbations of nonlinear Schrödinger equations: A Melnikov approach. Physica
D: Nonlinear Phenomena 2004;197(3-4):269-85.
Thank you

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Relevance of quantum mechanics in bacterial

  • 1. Relevance of Quantum Mechanics in Bacterial Communication Credit Seminar I
  • 3. ◎ We think that a multiple of bacteria are stronger than a few and thus by union are able to overcome obstacles too great for the few. ◎ There are many situations when the bacterial population behave co-operatively and recognize self and non-self which can be highly advantageous. ◎ In the contexts of symbiosis, niche adaptation, production of secondary metabolites and for facilitating population migration. ◎ These bacterial communities are glued together by EPS, which is called a biofilm.
  • 6. ◎ Autoinducers are wide variety of molecules, which are used by bacteria for inter- species and intra- species communication process. ◎ When a threshold concentration of the signaling molecules is achieved, a coordinated change in bacterial collective behavior is initiated. ◎ This bacterial collective behavior is known as quorum sensing by which bacterial cell integrate in order to determine their optimal survival strategy.
  • 7. ◎ Characteristic features to trigger the networking system- ◉ The production of the quorum sensing signal takes place during specific stages of growth, under certain physiological conditions, or in accordance to environmental changes. ◉ The quorum sensing signal diffuses in the extracellular space and is recognized by specific bacterial receptors.
  • 8. ◉ The accumulation of a critical threshold concentration of the quorum sensing signal initiates a firm response. ◉ The cellular response further offer beyond physiological changes that metabolize or detoxify the molecule.
  • 9. QUORUM SENSING ◎ Quorum sensing coordinates the gene expression, when the bacterial cell population has reached a high cell density. ◎ Quorum sensing is responsible for mediating a variety of social activities in biofilms, which include the swarming motility, biofilm dispersion, biofilm growth and antimicrobial resistance etc. ◎ It is found that quorum sensing regulating EPS produced during biofilm formation.
  • 10. ◎ Bacterial coordinated phenotypes are driven via quorum sensing systems, beneficial only at certain cell densities. ◎ Quorum sensing can function as a way to outcompete neighbors in patches occupied by many different genotypes. ◎ In these communities, competing microbial genotypes gradually segregated over time leading to positive correlation between density and genetic similarity between neighboring cells.
  • 11. ◎ Growing aggressively made quorum sensing genotypes a match for competitors. ◎ During growth, cells secrete autoinducers, that accumulate in the environment and high autoinducer concentration around cells induces expression as well as promotes metabolical traits. ◎ Cells can tune the expression of density-dependent phenotypes, like virulence factors or secreted enzymes.
  • 12. Quorum sensing Gram negative Acyl homoserine lactone (AHL) and quinolone Gram positive autoinducer peptide (AIP), ranges 5-34 amino acids in length
  • 14. ◎ The electrical communication within bacterial communities occurs through spatially propagating waves of potassium. ◎ When these communities grow larger, the supply of nutrient to the interior cells becomes limited because the nutrient consumption is increasing. ◎ At the same time, this nutrient consumption is associated with the growth of multiple layers of cells in the biofilm periphery.
  • 15. ◎ The conflict between starvation and protection is resolved through emergence of long range metabolic co- dependence between interior and peripheral cells and gives rise to collective oscillations within biofilms. ◎ Synchronized collective oscillations of membrane potential and the bacterial ion channels activate the electrical communication.
  • 16. ◎ In 2017, it has been discovered that two B. subtilis biofilm communities undergoing metabolic oscillations become coupled through electrical signaling and synchronize their growth dynamics. ◎ It is confirmed that biofilms resolve this conflict by switching from in phase to anti- phase. Different biofilm communities take turns consuming nutrients. ◎ Thus distant biofilms can coordinate their behavior to resolve nutrient competition through time-sharing.
  • 18. ◎ In biology, a vast range of intercellular coupling mechanisms lead to synchronized oscillators which govern fundamental physiological processes such as cardiac function, respiration, insulin secretion, and circadian rhythms. ◎ Synthetic biology can be broadly parsed into efforts aimed at the large- scale synthesis of DNA and the forward engineering of genetic circuits from known biological components ◎ In the area of DNA synthesis, pathways have been perturbed and replaced in an effort to understand the network motifs and transcriptional regulatory mechanisms that control cellular processes and elicit phenotypic responses ◎ Here, the original toggle switch and oscillator have inspired the design and construction of circuits capable of controlling cellular population growth, generating specific patterns, triggering biofilm development, shaping intracellular noise, detecting edges in an image, and counting discrete cellular events.
  • 19. ◎ A unifying theme for most of the genetic circuit studies is a particular focus on dynamical behavior. Thus the circuits are constructed and monitored in single cells, typically with fluorescent reporters, and new measurement technologies are often developed in parallel. ◎ Tools from the fields of nonlinear dynamics and statistical physics are extremely useful in both the generation of design specifications and for careful comparison between experiment and computational model. ◎ The synchronized oscillator design is based on elements of the quorum sensing machineries in Vibrio fisheri and Bacillus Thurigensis. The luxI (from V. fischeri), aiiA (from B. Thurigensis) and yemGFP genes are placed under the control of three identical copies of the luxI promoter.
  • 20.
  • 21. CHEMICAL METHODS TO INTERROGATE BACTERIAL QUORUM SENSING PATHWAYS
  • 22. ◎ QS challenges the traditional notion of bacteria as autonomous agents by permitting them to function as multicellular groups and thrive in specific environmental niches. ◎ Many of the phenotypes can have significant impacts on human health, agricultural yields, industrial manufacturing, and ecology. ◎ Binding of the AIs to their target receptors activates the transcription of genes required for QS phenotypes, along with those associated with AI biosynthesis. ◎ Increased production of the AI signal once a quorum is reached enhances the sensitivity of the signaling process and facilitates population-wide synchronization of the QS-regulated phenotype.
  • 23. Combinatorial approaches • AHL analogues on polystyrene resins • AHL analogues using small molecule microarray • DKP using small molecule microarray • AIP analogues on polystyrene resins • PQS analogues Quorum Quenching Antibodies • Designing antibodies to sequester AIs • Catalytic antibodies to inhibit QS Abiotic polymers for QS modulation • Polymer “Sinks” for AHL • Polymer “pools” for the controlled release of AIs Electrochemical techniques to study QS pyocyanin an electrochemical probe PQS as an electrochemical probe
  • 25. ◎ The densely packed bacterial populations develop a coordinated motion on the scales length 10µm to 100µm in comparison to the size of each single bacterium of order 3µm when the bacterial cell density reaches a sufficiently high value. ◎ The finite size of the bacteria indicates the existence of an intermediate length scale, which leads us to introduce a source of fluctuation, which is quite different than thermodynamic fluctuation. ◎ The swimming induced stresses on the bacteria that can change the local arrangement of bacteria induce stress fluctuations.
  • 26. ◎ Two different type of noise are present in the bacterial communication system and dominance of one over the other depends on the force ◎ F=f/ρg ◉ Where, o ρ = density ○ f = the volume of the forcing ○ g = the acceleration due to gravity
  • 27. VISCOSITY AND NON-LOCAL THEORY ◎ The rearrangement of the configuration of the coarse grained systems produce a noise that gives rise to kinematic viscosity. ◎ Noisy burger equation:- where, u= field v= kinematic viscosity ɳ= noise
  • 28. ◎ The Noisy burger equation gives the view of an internal structure of the complex biological communication system and viscosity is the property which makes the bacterial cells stick together into clusters predicted by Zeldovich approximation, just mimicking gravitational effect on the smaller scales. ◎ This approximation can describe the general structure of this nonlinear biological phenomenon. It is to be mentioned that the origin of viscosity is associated with the weakly non-local effects in the internal structure of the system.
  • 29. ◎ The kinematic viscosity plays a vital role in forming the metastable states of the bacteria responsible for quorum sensing. ◎ Kwak Transformation is a reaction- diffusion system, which gives the mathematical framework for the pattern formation. Where,
  • 31. ◎ In this multicellular system bacterial cells form different patterns based on chemical gradients of QSM signal that is synthesized by quorum sensing bacterial cells. ◎ Furthermore, the mathematical approach is able to predict how the system behaves if we change the initial value. ◎ It was observed that the quorum takes place in a certain range of kinematic viscosity([0.01, 0.32]m /s), which is considered as very small viscosity of the fluid. ◎ The behaviour changes with the initial data and system forms different wave patterns. 2
  • 33. ◎ The potassium ions follow non-linear Schrödinger equation. Where, ᴪ= wave function of Potassium ion. i = imaginary unit ħ= reduced Planck constant (1.054*10-34 J.s) r= position vector t= time Ĥ= Hamiltonian operator ◎ This non-linear Schrödinger equation is valid at the level of ion channel where as the perturbation becomes predominant at the cellular level.
  • 34. ◎ This fluctuation gives rise to the perturbation on non-linear Schrödinger equation and we get generalized Complex Ginzburg- Landau(GL) equation. Where, F= free energy Fn= free energy in normal phase α & β= phenomenological parameters m= effective mass e= charge of an electron A= magnetic vector potential B= magnetic field µ0= refractive index ◎ This Complex GL equation is used for the description of cellular communication through the chemical molecules and also needed to understand the generation of various patterns in Biofilms.
  • 36. ◎ One can find the solution may narrate biochemical systems on a continuum where the binary categorization of alive not alive is replaced by states that are increasingly lifelike. ◎ Following this path, each iteration of constructing an artificial cell could be objectively and quantifiably evaluated in terms of likeness to a target natural cell. ◎ Recent experiment has attempted to reconstruct the well characterized quorum sensing pathways to build an artficial cell which can mimic the ability of the natural cells of V. fisheri, E. coli, P. aeruginosa.
  • 37. ◎ Turing (1950), in his seminal paper pointed out that the ability of a machine to deceive a judge through textual communication into believing that the machine is a person was used to circumvent the problem of defining intelligence. ◎ A cellular Turing test is possible because all cells can communicate from quorum sensing pathways in bacteria to pheromone responses in higher organism. ◎ Moreover these artificial cells containing DNA supports the transcription and translation prosesses which can express genes.
  • 38. ◎ The genetic constructs in water-in –oil emulsion droplets are able to either sense and send quorum sensing molecules. ◎ Since quorum sensing is directly linked to gene expression, the next gereration sequencing technology can be used to quantifiably evaluate the extent of mimicry. ◎ A futher investigation is required for a much deeper understanding of life and the cellular Turing test can be a helful guide to achive this goal
  • 40. ◎ Bacteria use chemical signaling molecules, which are called as quorum sensing molecules (QSMs) or autoinducers. ◎ The ion channels in bacteria conduct a long-range electrical signaling within biofilm communities through propagated waves of potassium ions and biofilms attracts other bacterial species too. ◎ The bacterial communication mechanism is explained by complex Ginzburg- Landau equation and the formation of patterns depends on kinematic viscosity associated with internal noise. ◎ The potassium wave propagation is described by the non- linear Schrödinger equation.
  • 41. ◎ By adding perturbation to nonlinear Schrödinger equation one arrives at Complex Ginzburg-Landau equation which is used to understand the bacterial communication as well as pattern formation in Biofilms for certain range of kinematic viscosity. ◎ Here, the perturbation is due to the existence of non- thermal fluctuations associated to the finite size of the bacteria. ◎ It sheds new light on the relevance of quantum formalism in understanding the cell-to-cell communication.
  • 43. ◎ Majumdar S, Roy S. Relevance of quantum mechanics in bacterial communication. NeuroQuantology 2018 Jan 29;16(3). ◎ Majumdar S, Pal S. Bacterial intelligence: imitation games, time-sharing, and long- range quantum coherence. Journal of Cell Communication and Signaling. 2017;11(3):281-84. ◎ Majumdar S, Roy S, Llinas R. Bacterial Conversations and Pattern Formation. bioRxiv 2017:098053. ◎ Majumdar S, Mondal S. Conversation game: talking bacteria. Journal of Cell Communication and Signaling 2016;10(4):331-35. ◎ Roy S, Llinas R. Non-local hydrodynamics of swimming bacteria and self-activated process. InBIOMAT 2015: International Symposium on Mathematical and Computational Biology 2016:153-65. ◎ Roy S, Llinás R. Relevance of quantum mechanics on some aspects of ion channel function. Comptes Rendus Biologies 2009;332(6):517-22. ◎ Ván P, Fülöp T. Weakly non-local fluid mechanics: the Schrödinger equation. InProceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 2006; 462 (2066):541-57. ◎ Ribeiro AL, de Faria JP. Weakly nonlocal hydrodynamics and the origin of viscosity in the adhesion model. Physical Review D 2005;71(6):067302. ◎ Cruz-Pacheco G, Levermore CD, Luce BP. Complex Ginzburg– Landau equations as perturbations of nonlinear Schrödinger equations: A Melnikov approach. Physica D: Nonlinear Phenomena 2004;197(3-4):269-85.