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Mathematical model of talking bacteria

Sarangam Majumdar

Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 1 / 36

Overview

Quorum sensing

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Overview

Quorum sensing Mathematical Model

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Overview

Quorum sensing Mathematical Model Discussion

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Overview

Quorum sensing Mathematical Model Discussion Observation

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Overview

Quorum sensing Mathematical Model Discussion Observation Forced Burger equation, Kawak transformation and Reaction diffusion system

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Overview

Quorum sensing Mathematical Model Discussion Observation Forced Burger equation, Kawak transformation and Reaction diffusion system pattern formation

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Overview

Quorum sensing Mathematical Model Discussion Observation Forced Burger equation, Kawak transformation and Reaction diffusion system pattern formation Quantum Perspective

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 2 / 36

Quorum sensing

Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 3 / 36

Quorum sensing

Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 4 / 36

Quorum sensing

Batch Culture of Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 5 / 36

Quorum sensing

Quorum Sensing A co-ordinated change in bacterial behavior depending on the concentration of the autoinducers (the signalling molecules) for facilitating bacterial adaptation to environmental stress. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 6 / 36

Quorum sensing (Quorum sensing molecules (QSM) used by different kind of bacteria)

Signal Organisms C4-HSL (an AHL) Aeromonas hydrophila, Pseudomonas aeruginosa C6-HSL Erwinia carotovora,Pseudomonas aureofaciens, Yersinia enterocolitica 3-Oxo-C6-HSL

• E. carotovora, Vibrio fischeri, Y. enterocolitica

3-Oxo-C8-HSL Agrobacterium Tumefaciens Autoinducing Peptide (AIP)-I Straphylococcus aureus Group I strains AI-2 (S-THMF-borate) Vibrio harveyi Farnesol Candida albicans Structure of C4-HSL Structure of C6-HSL Structure of AI-2 (S-THMF-borate) Structure of 3-Oxo-C6-HSL Structure of Autoinducing Peptide (AIP)-I Structure of Farnesol Structure of 3-Oxo-C8-HSL Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 7 / 36

Mathematical Model

The Gelf’ and triple

Let V and H be two Hilbert spaces on C, with V ⊂ H, V dense in H, the canonical injection of V into H being continuous. We denote (., .) the inner product in H, |.| its associated norm and ||.|| the norm in V. By Riesz Theorem, to each bounded antilinear form on H we can associate a unique element u belong to H such that this form a map v → (u, v) from H to C reciprocally,an element u ∈ H defines as a bounded antilinear map on H. Thus identify H to its antidual H

′. The space H ≡ H ′ is identified to the subspace of V ′, the

antidual space to V. We get V ⊂ H ≡ H

′ ⊂ V ′

Moreover, H is dense and continuously embedded into V

′ and we can easily use the

same notation for the inner product in H and for the duality between V

′ and V.

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 8 / 36

Mathematical Model

Resolvent Estimates

Theorem

Let A be an mα- accretive operator. Then ∀z / ∈ Sα we have the estimates ||(zI − A)−1||H→H ≤ 1 d(z, Sα) ||A(zI − A)−1||H→H ≤ |z| d(z, Sα) Moreover, the maps z → (zI − A)−1 from Sc

α to L(H, H) and L(H, D(A)) are

continuous and infinitely differentiable (in the sense of C). Conversely if we assume that ∀z / ∈ Sα ,zI − A is an isomorphism from D(A) to H and that, ∀z = 0 with |argz| = α + π 2 ,||(zI − A)−1||H→H ≤ 1 |z| then A is mα - accretive.

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 9 / 36

Mathematical Model

Functions of operators : Analytic semigroups A → mα - accretive operator on H; r → a rational fraction bounded on Sα such that r(z) = r(∞) +

• j

rj (αj − zmj) with αj / ∈ Sα, mj ∈ ℵ∗ Defining operator r(A) by r(A) = r(∞)I +

• j

rj((αjI − A)−1)mj

Theorem

Let α ∈ [0, π 2 ]. There exists a constant 1 ≤ Cα ≤ 2 + 2 √ 3 such that for all mα - accretive operator A and for all rational fraction r bounded on the sector Sα, we have ||r(A)||H→H ≤ Cα sup

z∈Sα

|r(z)| Moreover, when α = π 2 , we have C π

2 = 1.

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 10 / 36

Mathematical Model

Corollary

When the function f is a uniform limit of rational fractions rn on Sα, the relation f(A) = lim

n→∞rn(A) defines an operator f(A) ∈ L(H, H) and it follows that

||f(A)||H→H ≤ Cα supz∈Sα |f(z)|

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 11 / 36

Mathematical Model

Corollary

When the function f is a uniform limit of rational fractions rn on Sα, the relation f(A) = lim

n→∞rn(A) defines an operator f(A) ∈ L(H, H) and it follows that

||f(A)||H→H ≤ Cα supz∈Sα |f(z)|

Lemma

For all α ∈ [0, π

2 ) we have

∀n ≥ 1, sup

z∈Sα

|e−z − ( 1 1 + z/n)n| ≤ 6 n cos2 α

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 11 / 36

Mathematical Model

Corollary

When the function f is a uniform limit of rational fractions rn on Sα, the relation f(A) = lim

n→∞rn(A) defines an operator f(A) ∈ L(H, H) and it follows that

||f(A)||H→H ≤ Cα supz∈Sα |f(z)|

Lemma

For all α ∈ [0, π

2 ) we have

∀n ≥ 1, sup

z∈Sα

|e−z − ( 1 1 + z/n)n| ≤ 6 n cos2 α

Corollary

Let α and β satisfy 0 ≤ α < α + β < π 2 . Then ∀t ∈ Sβ the function E(t) = exp(−tA) is well defined. Moreover E(t) ∈ L(H, H) and ||E(t)||H→H ≤ 1. RemarkThis corollary is valid in particular with t = 0 and we get E(0) = I.

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 11 / 36

Mathematical Model

Theorem

Let α ∈ [0, π 2 ). The family of operators E(t), t ≥ 0 satisfies the following properties ∀t, s ≥ 0, E(t + s) = E(t)E(s) ∀t ≥ 0, ||E(t)||H→H ≤ 1 ∀u0 ∈ H, the map t → E(t)u0 is continuous from ℜ+ to H. We say that this family is a semigroup of contractions stronly continuous on H and that the operator A is the infinitesimal generator of this semigroup. Remark: The theorem is still valid for α = π

2 . It is also valid ∀t, s ∈ S π 2 −α

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 12 / 36

Mathematical Model

Theorem

Let α ∈ [0, π 2 ). The family of operators E(t), t ≥ 0 satisfies the following properties ∀t, s ≥ 0, E(t + s) = E(t)E(s) ∀t ≥ 0, ||E(t)||H→H ≤ 1 ∀u0 ∈ H, the map t → E(t)u0 is continuous from ℜ+ to H. We say that this family is a semigroup of contractions stronly continuous on H and that the operator A is the infinitesimal generator of this semigroup. Remark: The theorem is still valid for α = π

2 . It is also valid ∀t, s ∈ S π 2 −α

Theorem

For all t ∈ Sβ with t = 0 and 0 ≤ α < α + β < π

2 , and ∀k ≥ 0, the operator E(t) ∈ L(H, D(Ak )). Furthermore, the map

t → E(t) from Sβ to L(H, H) is differentiable (holomorphic), and we have the estimate ∀t > 0, ||E(k)(t)||H→H = ||Ak E(t)||H→H ≤ k! 1 tk cosk α Remark: The property in Sβ is stated the semigroup E(.) is analytic (also holomorphic) Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 12 / 36

Evolution problem applied to Quorum sensing (Mathematical Model)

The mathematical model of the quorum sensing system as follows (u

′

(t), v) + a(u(t), v) = ∀ t > 0, ∀v ∈ V u(0) = u0 u is the concentration (external) of QSM. ρ is consider as mass density. a is diffusion coefficient. b = ϕǫ, where ϕ is assumed as average velocity of the QSM and ǫ is known as porosity ( ratio of liquid volume to the volume). R = cu is the source of the QSM concentration. R > 0 mean the quorum sensing is switch on. u ∈ C1((0, ∞); V)C0([0, ∞); H). The mathematical model of the quorum sensing in terms of a parabolic partial differential equation ρ ∂u ∂t − div(a∇u) + b.∇u + cu = fort > 0, x ∈ Ω u(t, x) = for t > 0, x ∈ ∂Ω u(0, x) = u0(x) for x ∈ Ω Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 13 / 36

Numerical Schemes: Space and time approximations

Time Approximation: Semi-Discretization in Time (P)

• u′(t) + Au(t)

= 0, ∀t > 0 u(0) = u0 Scheme 1. un+1−un

∆t

+ Aun = 0, (explicit) Euler method, Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 14 / 36

Numerical Schemes: Space and time approximations

Time Approximation: Semi-Discretization in Time (P)

• u′(t) + Au(t)

= 0, ∀t > 0 u(0) = u0 Scheme 1. un+1−un

∆t

+ Aun = 0, (explicit) Euler method, Scheme 2. un+1−un

∆t

+ Aun+1 = 0, (implicit) Euler method, Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 14 / 36

Numerical Schemes: Space and time approximations

Time Approximation: Semi-Discretization in Time (P)

• u′(t) + Au(t)

= 0, ∀t > 0 u(0) = u0 Scheme 1. un+1−un

∆t

+ Aun = 0, (explicit) Euler method, Scheme 2. un+1−un

∆t

+ Aun+1 = 0, (implicit) Euler method, Scheme 3. un+1−un

∆t

+ A un+un+1

2

= 0, Crank-Nicolson method, Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 14 / 36

Numerical Schemes: Space and time approximations

Time Approximation: Semi-Discretization in Time (P)

• u′(t) + Au(t)

= 0, ∀t > 0 u(0) = u0 Scheme 1. un+1−un

∆t

+ Aun = 0, (explicit) Euler method, Scheme 2. un+1−un

∆t

+ Aun+1 = 0, (implicit) Euler method, Scheme 3. un+1−un

∆t

+ A un+un+1

2

= 0, Crank-Nicolson method, Scheme 4. un+1 = r(∆tA)un, Runge-Kutta schemes, Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 14 / 36

Numerical Schemes: Space and time approximations

Time Approximation: Semi-Discretization in Time (P)

• u′(t) + Au(t)

= 0, ∀t > 0 u(0) = u0 Scheme 1. un+1−un

∆t

+ Aun = 0, (explicit) Euler method, Scheme 2. un+1−un

∆t

+ Aun+1 = 0, (implicit) Euler method, Scheme 3. un+1−un

∆t

+ A un+un+1

2

= 0, Crank-Nicolson method, Scheme 4. un+1 = r(∆tA)un, Runge-Kutta schemes,

Theorem

The operator A is mα-accretive, 0 ≤ α ≤ π

2 ,

r is an approximation of order p, and r is A(α)-acceptable. Then there exists a constants K such that ∀u0 ∈ D(Ap), the solution of scheme (4) satisfies ∀n ≥ 2, |u(tn) − un| ≤ K cosα ∆tp|Apu0| If in addition the rational fraction r is strongly A(α)-acceptable, we have for all u0 ∈ H, ∀n ≥ 2, |u(tn) − un| ≤ K ′ tp

n (cosα)p+1 ∆tp|u0|

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 14 / 36

Numerical Schemes: Space and time approximations

Hilbertian Integrals: Operator Integrals

Let Γ be an oriented simple curve in the complex plane and we consider the following maps z → f(z) z → B(z) Γ → H Γ → L(H, H)

• Γ f(z)dz is well defined if, ∀v ∈ H,
• Γ(f(z), v)dz is well defined and ∃ an element I2 ∈ L(H, H) such that

∀v ∈ H, (I1, v) =

• Γ(f(z), v)dz, Then we say

I1 =

• `

f(z)dz

• Γ B(z)dz is well defined if, ∀u and v ∈ H,
• Γ(B(z)u, v)dz is well defined and ∃ an element I2 ∈ L(H, H) such that

∀u, v ∈ H, (I2u, v) =

• Γ(B(z)u, v)dz, Then we say

I2 =

• `

B(z)dz A sufficient condition for I1 and I2 to be well defined

• f be continious on Γ with values in H and that
• Γ |f(z)||dz| < +∞
• B be continious on Γ with values in L(H, H) and that
• Γ ||B(z)||H→H|dz| < +∞.

Remark: Let us we assume that in a neighborhood of the curve Γ, ∀u, v ∈ H, z → (f(z), v) and z → (B(z)u, v) are

• holomorphic. Then,
• Γ

f(z)dz =

• Γ′ f(z)dz

and

• Γ

B(z)dz =

• Γ

B(z)dz for every holomorphic deformation Γ′ of Γ in this neighbourhood and with the same end-points. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 15 / 36

Numerical Schemes: Space and time approximations

Lemma Assume that A is an isomorphism from D(A) onto H and that A is mα-accretive, α ∈ [0, π

2 ]; let β ∈ (0, π − α). Then

∀λ / ∈ Sα+β, (λI − A)−1 = 1 2πi

• Γα+β

1 λ − z (zI − A)−1dz, where Γα+β denotes the counterclockwise oriented boundary of the sector Sα+β. Theorem Assume that A is an isomorphism from D(A) onto H and that A is mα-accretive, α ∈ [0, π

2 ]. Let β ∈ (0, π − α) and let f be a holomorphic function in the interior of

Sα+β, continious on Sα+β, and such that f(z) = 0(z−ǫ) when |z| → ∞ for some ǫ > 0. Then the operator f(A) ∈ L(H, H) is well defined and f(A) = 1 2πi

• Γα+β

f(z)(zI − A)−1dz.

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 16 / 36

Numerical Schemes: Space and time approximations

Space Approximation: Finite Element Semidixcretization

we consider the problem in the following way (Ph)      find uh ∈ C1([0, ∞); Vh) such that ∀t > 0, ∀wh ∈ Vh, (u′

h(t), wh) + a(uh(t), wh) = 0,

uh(0) = u0h, given in Vh. Introducing Ah ∈ (Vh, Vh) defined by ∀vh, wh ∈ Vh, Ahvh ∈ Vh and (Ahvh, wh) = a(vh, wh), (Ph)

• ∀t > 0,

u′

h(t) + Ahuh(t) = 0,

uh(0) = u0h. Remark: unique solution:uh(t) = Eh(t)u0h Eh(t) := exp(−tAh) = I − tAh + · · · + (−t)k k! Ak

h + · · · =

1 2πi

• Γα+β

e−tz(zI − Ah)−1dz . Remark: The operator Ah is mα-accretive on the space Vh endowed with the inner product of H. Since we are in finite dimension, the operator Ah is necessarily bounded D(Ah) = Vh. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 17 / 36

Numerical Schemes: Space and time approximations

Space Approximation: Finite Element Semidixcretization

Theorem

Assume (H1) and u0 ∈ H. Then the following error estimate holds ∀t > 0, |u(t) − uh(t)| ≤ |Phu0 − u0h| + Ch2 tcos3α|u0| Moreover ∀t > 0, tk|u(K)(t) − u(k)

h (t)| ≤ (

k! coskα|Phu0 − u0h| + Ch2 tcosk+3α|u0|). with the choice of uoh = Phu0, this provides an error 0(h2/t)

Theorem

Assume u0 ∈ D(A) and (H1), then ∀t > 0, |u(t) − uh(t)| ≤ |u0 − u0h| + Ch2 cosα|Au0|

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 18 / 36

Numerical Schemes: Space and time approximations

Space Semi-Discretizations: Higher Order Methods

we introduce the following functions. ϕs(z) = zs (1 + z)s , ψs(z) = 1 (1 + z)s , where z ∈ Sα, 0 ≤ s < 1.

Definitions

For 0 ≤ s < 1 and k ∈ N D(As) := {u ∈ H; ∃v ∈ H with u = ψs(A)v} if u ∈ D(As), Asu = ϕs(A)v D(Ak+s) := {u ∈ D(Ak ); Ak u ∈ D(As)} if u ∈ D(Ak+s), Ak+su = As(Ak u)

Theorem

a) If 0 ≤ s ≤ s + s′, D(As+s′ ) ⊂ D(As) ∩ D(As′ ). Besides, if u ∈ D(As+s′ ), then As′ u ∈ D(As) and As+s′ u = As(As′ u). b) If u0 ∈ D(As), then, ∀t ≥ 0, E(t)u0 ∈ D(As) and we have AsE(t)u0 = E(t)Asu0. c)For s ∈ [0, 1], the operator As is m-sα-accretive. d)When A proceeds from a bounded V -elliptic hermitic form, that is, when A = A∗ and ∀v ∈ V, a0||v||2 ≤ (Av, v) ≤ M||v||2, then D(A1/2) = V and ∀v ∈ V, a0||v|| ≤ |A1/2v| ≤ √ M||v|| Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 19 / 36

Numerical Schemes: Space and time approximations

Theorem Let k ≥ 1 be an interger. Under the assumptions u0 ∈ D(A(k+1)/2) and (Hk ) we have ∀t > 0, |u(t) − uh(t)| ≤ |u0 − u0h| + Chk+1 cosα |A(k+1)/2u0| Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 20 / 36

Numerical Schemes: Space and time approximations

Theorem Let k ≥ 1 be an interger. Under the assumptions u0 ∈ D(A(k+1)/2) and (Hk ) we have ∀t > 0, |u(t) − uh(t)| ≤ |u0 − u0h| + Chk+1 cosα |A(k+1)/2u0| Lemma Let k ≥ 1 be an integer. Conditions (νk ) and (ν1) imply (Hk ), Conditions (ν1) and (ν2) imply ∀f ∈ H, ||Th(T − Th)f|| ≤ Ch3|f|, Conditions (ν1) and (ν3) imply ∀f ∈ H, |Th(T − Th)f| ≤ Ch4|f| Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 20 / 36

Numerical Schemes: Space and time approximations

Theorem Let k ≥ 1 be an interger. Under the assumptions u0 ∈ D(A(k+1)/2) and (Hk ) we have ∀t > 0, |u(t) − uh(t)| ≤ |u0 − u0h| + Chk+1 cosα |A(k+1)/2u0| Lemma Let k ≥ 1 be an integer. Conditions (νk ) and (ν1) imply (Hk ), Conditions (ν1) and (ν2) imply ∀f ∈ H, ||Th(T − Th)f|| ≤ Ch3|f|, Conditions (ν1) and (ν3) imply ∀f ∈ H, |Th(T − Th)f| ≤ Ch4|f| Theorem We assume u0 ∈ H, (ν1) and (νk ), k=1, 2 or 3. For k = 2 we also suppose that ∀vh ∈ Vh, |A1/2

h

vh| ≤ K||vh|| Then we have the bound ∀t > 0, |u(t) − uh(t)| ≤ |Phu0 − u0h| +

Chk+1 t(k+1)/2cos(k+5)/2α |u0|

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 20 / 36

Numerical Schemes: Space and time approximations

Theorem Let k ≥ 1 be an interger. Under the assumptions u0 ∈ D(A(k+1)/2) and (Hk ) we have ∀t > 0, |u(t) − uh(t)| ≤ |u0 − u0h| + Chk+1 cosα |A(k+1)/2u0| Lemma Let k ≥ 1 be an integer. Conditions (νk ) and (ν1) imply (Hk ), Conditions (ν1) and (ν2) imply ∀f ∈ H, ||Th(T − Th)f|| ≤ Ch3|f|, Conditions (ν1) and (ν3) imply ∀f ∈ H, |Th(T − Th)f| ≤ Ch4|f| Theorem We assume u0 ∈ H, (ν1) and (νk ), k=1, 2 or 3. For k = 2 we also suppose that ∀vh ∈ Vh, |A1/2

h

vh| ≤ K||vh|| Then we have the bound ∀t > 0, |u(t) − uh(t)| ≤ |Phu0 − u0h| +

Chk+1 t(k+1)/2cos(k+5)/2α |u0|

Lemma We suppose that the operator Ah proceeds from a bounded V-elliptic sesquilinear form and also that there exists a constants C such that ∀vh ∈ Vh |(Ah − A∗

h )vh| ≤ C||vh||.

Then, there exists a constant K such this condition holds Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 20 / 36

Numerical Schemes: Space and time approximations

Full Discretization

(Pap)

• u0

h

= u0h, un+1

h

= r(∆tAh)un

h,

n ≥ 0 Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 21 / 36

Numerical Schemes: Space and time approximations

Full Discretization

(Pap)

• u0

h

= u0h, un+1

h

= r(∆tAh)un

h,

n ≥ 0 Theorem Assume that the hypothesis “u0 ∈ D(A), (H1), r is A(α)-acceptable and of order 1" hold. then ∀n ≥ 1, |u(tn) − un

h| ≤ C(|u0 − u0h| +

h2 + ∆t cos α |Au0|) Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 21 / 36

Numerical Schemes: Space and time approximations

Full Discretization

(Pap)

• u0

h

= u0h, un+1

h

= r(∆tAh)un

h,

n ≥ 0 Theorem Assume that the hypothesis “u0 ∈ D(A), (H1), r is A(α)-acceptable and of order 1" hold. then ∀n ≥ 1, |u(tn) − un

h| ≤ C(|u0 − u0h| +

h2 + ∆t cos α |Au0|) Theorem Assume that the hypothesis “u0 ∈ D(A2), (ν1), (νk ), k = 1, 2 or 3, r is A(α)-acceptable and of order 2" hold. Then ∀n ≥ 1, |u(tn) − un

h| ≤ C(|u0 − u0h| +

hk+1 + ∆t2 cos α |A2u0|) If in addition the rational fraction r is strongly A(α)-acceptable and of order p ≥ 2, we have ∀n ≥ 1, |u(tn) − un

h| ≤ C(|u0 − u0h| +

hk+1 cos α |A2u0| + ∆tp tp−2

n

cosp−1 α |A2u0|) Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 21 / 36

Evolution problem applied to Quorum sensing (Numerical Euler Scheme)

Euler-Forward Method Euler-Backward Method Modified-Euler Method Euler Method Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 22 / 36

Evolution problem applied to Quorum sensing (Runge-Kutta Scheme)

Runge-Kutta Method Runge-Kutta 4 Method Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 23 / 36

Evolution problem applied to Quorum sensing (Finite Element Analysis)

Concentration graph using Finite element method log(N) vs log(error) (For n= 10)concentration behaviour using FEM (For n= 100)concentration behaviour using FEM Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 24 / 36

Evolution problem applied to Quorum sensing (Finite Element analysis)

(For n= 1000)concentration behaviour using FEM (For n= 10000)concentration behaviour using FEM Number of Nodes(n) Peclet Number 10 4.5455 100 4.950 1000 0.0500 10000 0.0050 Peclet numbers with different nodes Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 25 / 36

Evolution problem applied to Quorum sensing (Numerical Experiments)

Concentration graph of experiment

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 26 / 36

Evolution problem applied to Quorum sensing (Comparison between the experiment data and the numerical result of the

concentration behaviour )

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 27 / 36

Observation

We proposed a mathematical model of the quorum sensing mechanism which is basically a convection - diffusion model. The model predicts the behaviour of the QSM concentration in space and time. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 28 / 36

Observation

We proposed a mathematical model of the quorum sensing mechanism which is basically a convection - diffusion model. The model predicts the behaviour of the QSM concentration in space and time. We observed a negative diffusion coefficient which occurs in this complex biochemical phenomenon. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 28 / 36

Observation

We proposed a mathematical model of the quorum sensing mechanism which is basically a convection - diffusion model. The model predicts the behaviour of the QSM concentration in space and time. We observed a negative diffusion coefficient which occurs in this complex biochemical phenomenon. We used various numerical schemes (explicit Euler, Implicit Euler, explicit Runge-Kutta, implicit Runge-Kutta, Finite element method) and compared the result in sense of approx- imation of the solution and the stability of methods. We

• bserved that finite element approximation gives a better result than others. Moreover, Runge - Kutta scheme also gives

a A-stability and con- centration curves in time which is more accurate with the experimental data. The batch culture of the quorum sensing system and our proposed model approximately give a good result. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 28 / 36

Observation

We proposed a mathematical model of the quorum sensing mechanism which is basically a convection - diffusion model. The model predicts the behaviour of the QSM concentration in space and time. We observed a negative diffusion coefficient which occurs in this complex biochemical phenomenon. We used various numerical schemes (explicit Euler, Implicit Euler, explicit Runge-Kutta, implicit Runge-Kutta, Finite element method) and compared the result in sense of approx- imation of the solution and the stability of methods. We

• bserved that finite element approximation gives a better result than others. Moreover, Runge - Kutta scheme also gives

a A-stability and con- centration curves in time which is more accurate with the experimental data. The batch culture of the quorum sensing system and our proposed model approximately give a good result. Finally, we do some numerical experiment with this density dependent behaviour of the bacterial talk with only convection effect which illustrate a behaviour of the quorum sensing in space and time. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 28 / 36

The dynamics of QS

Let us consider u(x, t) as the concentration of the cell-to -cell signalling which function as pheromones (also called as autoinducers) when they function in part to stimulate their own synthesis).This is the bacterial population means of determining its numerical size (or density). Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 29 / 36

The dynamics of QS

Let us consider u(x, t) as the concentration of the cell-to -cell signalling which function as pheromones (also called as autoinducers) when they function in part to stimulate their own synthesis).This is the bacterial population means of determining its numerical size (or density). If we now assume that ν is the viscosity of the modified fluid known as bacterial fluid or living fluid, this viscosity will then become a most important parameter for the long-time dynamics of the quorum sensing system. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 29 / 36

The dynamics of QS

Let us consider u(x, t) as the concentration of the cell-to -cell signalling which function as pheromones (also called as autoinducers) when they function in part to stimulate their own synthesis).This is the bacterial population means of determining its numerical size (or density). If we now assume that ν is the viscosity of the modified fluid known as bacterial fluid or living fluid, this viscosity will then become a most important parameter for the long-time dynamics of the quorum sensing system. In our approach, we consider F as a dissipative force (energy is lost from the quorum sensing system when motion takes place).The loss from the degrees of freedom is converted into radiation (the motion of new particles created by the motion -light usually, bioluminesence for the QS system). Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 29 / 36

The dynamics of QS

Let us consider u(x, t) as the concentration of the cell-to -cell signalling which function as pheromones (also called as autoinducers) when they function in part to stimulate their own synthesis).This is the bacterial population means of determining its numerical size (or density). If we now assume that ν is the viscosity of the modified fluid known as bacterial fluid or living fluid, this viscosity will then become a most important parameter for the long-time dynamics of the quorum sensing system. In our approach, we consider F as a dissipative force (energy is lost from the quorum sensing system when motion takes place).The loss from the degrees of freedom is converted into radiation (the motion of new particles created by the motion -light usually, bioluminesence for the QS system). At an increased velocities, the force of friction increases as a higher power of the relative velocity and the QSM travelling through modified fluids at high Reynolds number (Re = vL

ν ), where ν → viscosity of the fluid.

Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 29 / 36

The dynamics of QS

Let us consider u(x, t) as the concentration of the cell-to -cell signalling which function as pheromones (also called as autoinducers) when they function in part to stimulate their own synthesis).This is the bacterial population means of determining its numerical size (or density). If we now assume that ν is the viscosity of the modified fluid known as bacterial fluid or living fluid, this viscosity will then become a most important parameter for the long-time dynamics of the quorum sensing system. In our approach, we consider F as a dissipative force (energy is lost from the quorum sensing system when motion takes place).The loss from the degrees of freedom is converted into radiation (the motion of new particles created by the motion -light usually, bioluminesence for the QS system). At an increased velocities, the force of friction increases as a higher power of the relative velocity and the QSM travelling through modified fluids at high Reynolds number (Re = vL

ν ), where ν → viscosity of the fluid.

The rate of change of concentration of lux I per unit volume is explicitly dependent on the concentration of AHL through the activation of expression of the operon by such complex. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 29 / 36

The dynamics of QS

Here, we model the QS system as the force Burgers equation via ut + uux − νuxx = F(x). This forced Burgers equation can be transformed by the transformation J(u) = (u, ux , − 1

2 u2) with v = ux and w = − 1 2 u2 into a a reaction diffusion system as

ut = νuxx + wx + F(x) u(2x, t) = u(0, t) vt = νvxx + wxx + F ′(x) v(2x, t) = v(0, t) wt = νwxx + νv2 + u2v − uF(x) w(2x, t) = w(0, t) The given initial condition are specified as u(x, 0) = u0(x), v(x, 0) = v0(x), w(x, 0) = w0(x). Here, we use the Kwak transformation in a slightly different manner than originally used by Kwak. Thus, structured our model for the quorum sensing system as follow ut = uxx − uux + h(x) (1) With h(x) = F(x)

ν2

By setting u = u, v = ux and w = − 1

2 u2, we obtain the new system as

ut = uxx + wx + h(x) (2) vt = vxx + wxx + h′(x) (3) wt = wxx + v2 + u2v − uh(x) (4) Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 30 / 36

Pattern formation Quorum sensing

Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 31 / 36

Pattern formation Quorum sensing

Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 32 / 36

Pattern formation Quorum sensing

Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 33 / 36

Pattern formation Quorum sensing

Quorum Sensing Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 34 / 36

Quantum QS

Quorum Sensing Why bacteria evolve such signal integration circuits and what is the advantage of using more than one AIs since all signaling pathways merge in one? Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 35 / 36

Quantum QS

Quorum Sensing Why bacteria evolve such signal integration circuits and what is the advantage of using more than one AIs since all signaling pathways merge in one? It still remains unclear how crosstalk between C8HSL and 3OC6HSL affects the information that the bacterium obtains through quorum sensing. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 35 / 36

Quantum QS

Quorum Sensing Why bacteria evolve such signal integration circuits and what is the advantage of using more than one AIs since all signaling pathways merge in one? It still remains unclear how crosstalk between C8HSL and 3OC6HSL affects the information that the bacterium obtains through quorum sensing. Although QS has been extensively studied in well-mixed systems, the ability of diffusing QS signals to synchronize gene expression in spatially extended colonies is not well understood. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 35 / 36

Quantum QS

Quorum Sensing Why bacteria evolve such signal integration circuits and what is the advantage of using more than one AIs since all signaling pathways merge in one? It still remains unclear how crosstalk between C8HSL and 3OC6HSL affects the information that the bacterium obtains through quorum sensing. Although QS has been extensively studied in well-mixed systems, the ability of diffusing QS signals to synchronize gene expression in spatially extended colonies is not well understood. Origin of QS. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 35 / 36

References

1 Miller, Melissa B., and Bonnie L. Bassler. "Quorum sensing in bacteria."Annual Reviews in Microbiology 55.1 (2001): 165-199. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 36 / 36

References

1 Miller, Melissa B., and Bonnie L. Bassler. "Quorum sensing in bacteria."Annual Reviews in Microbiology 55.1 (2001): 165-199. 2 Majumdar, Sarangam, and Subhoshmita Mondal. "Conversation game: talking bacteria." Journal of Cell Communication and Signaling (2016): 1-5. (Springer) Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 36 / 36

References

1 Miller, Melissa B., and Bonnie L. Bassler. "Quorum sensing in bacteria."Annual Reviews in Microbiology 55.1 (2001): 165-199. 2 Majumdar, Sarangam, and Subhoshmita Mondal. "Conversation game: talking bacteria." Journal of Cell Communication and Signaling (2016): 1-5. (Springer) 3 Roy, Sisir, and Rodolfo Llinas. "Non-local hydrodynamics of swimming bacteria and self-activated process.", BIOMAT 2015 Proceedings of the International Symposium on Mathematical and Computational Biology. World Scientific (2016): 153-165. Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 36 / 36

References

1 Miller, Melissa B., and Bonnie L. Bassler. "Quorum sensing in bacteria."Annual Reviews in Microbiology 55.1 (2001): 165-199. 2 Majumdar, Sarangam, and Subhoshmita Mondal. "Conversation game: talking bacteria." Journal of Cell Communication and Signaling (2016): 1-5. (Springer) 3 Roy, Sisir, and Rodolfo Llinas. "Non-local hydrodynamics of swimming bacteria and self-activated process.", BIOMAT 2015 Proceedings of the International Symposium on Mathematical and Computational Biology. World Scientific (2016): 153-165. 4 Sarangam Majumdar, Sukla Pal. "Quorum sensing : a quantum perspective" Journal of Cell Communication and Signaling (2016): 1-3. (Springer) Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 36 / 36

References

1 Miller, Melissa B., and Bonnie L. Bassler. "Quorum sensing in bacteria."Annual Reviews in Microbiology 55.1 (2001): 165-199. 2 Majumdar, Sarangam, and Subhoshmita Mondal. "Conversation game: talking bacteria." Journal of Cell Communication and Signaling (2016): 1-5. (Springer) 3 Roy, Sisir, and Rodolfo Llinas. "Non-local hydrodynamics of swimming bacteria and self-activated process.", BIOMAT 2015 Proceedings of the International Symposium on Mathematical and Computational Biology. World Scientific (2016): 153-165. 4 Sarangam Majumdar, Sukla Pal. "Quorum sensing : a quantum perspective" Journal of Cell Communication and Signaling (2016): 1-3. (Springer) 5 Sarangam Majumdar, Sisir Roy, Rodolfo Llinas . " Bacterial Conversations and Pattern Formation" ( submitted). Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 36 / 36

References

1 Miller, Melissa B., and Bonnie L. Bassler. "Quorum sensing in bacteria."Annual Reviews in Microbiology 55.1 (2001): 165-199. 2 Majumdar, Sarangam, and Subhoshmita Mondal. "Conversation game: talking bacteria." Journal of Cell Communication and Signaling (2016): 1-5. (Springer) 3 Roy, Sisir, and Rodolfo Llinas. "Non-local hydrodynamics of swimming bacteria and self-activated process.", BIOMAT 2015 Proceedings of the International Symposium on Mathematical and Computational Biology. World Scientific (2016): 153-165. 4 Sarangam Majumdar, Sukla Pal. "Quorum sensing : a quantum perspective" Journal of Cell Communication and Signaling (2016): 1-3. (Springer) 5 Sarangam Majumdar, Sisir Roy, Rodolfo Llinas . " Bacterial Conversations and Pattern Formation" ( submitted). 6 Majumdar S, Classical and numerical foundation theory: parabolic evolution problems with application, Thesis (2016). Sarangam Majumdar (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita’ degli Studi dell’Aquila) Mathematical model of talking bacteria 36 / 36