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 AI-2 (S-THMF-borate) Structure of C4-HSL Structure of C6-HSL 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 ′ is identified to the subspace of V ′ . The space H ≡ H ′ , the identify H to its antidual H antidual space to V . We get ′ ⊂ V ′ V ⊂ H ≡ H ′ and we can easily use the Moreover, H is dense and continuously embedded into V ′ and V . same notation for the inner product in H and for the duality between 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 m α - accretive operator. Then ∀ z / ∈ S α we have the estimates 1 || ( zI − A ) − 1 || H → H ≤ d ( z , S α ) | z | || A ( zI − A ) − 1 || H → H ≤ d ( z , S α ) Moreover, the maps z → ( zI − A ) − 1 from S c 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 / ∈ S α , 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 m α - 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 → m α - accretive operator on H ; r → a rational fraction bounded on S α such that r j � ∈ S α , m j ∈ ℵ ∗ r ( z ) = r ( ∞ ) + with α j / ( α j − z mj ) j Defining operator r ( A ) by � r j (( α j I − A ) − 1 ) mj r ( A ) = r ( ∞ ) I + j Theorem Let α ∈ [ 0 , π 2 ] . There exists a constant 1 ≤ C α ≤ 2 + 2 √ such that for all m α - 3 accretive operator A and for all rational fraction r bounded on the sector S α , we have || r ( A ) || H → H ≤ C α sup | r ( z ) | z ∈ S α 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 r n on S α , the relation f ( A ) = lim n →∞ r n ( A ) defines an operator f ( A ) ∈ L ( H , H ) and it follows that || f ( A ) || H → H ≤ C α sup z ∈ S α | 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 r n on S α , the relation f ( A ) = lim n →∞ r n ( A ) defines an operator f ( A ) ∈ L ( H , H ) and it follows that || f ( A ) || H → H ≤ C α sup z ∈ S α | f ( z ) | Lemma For all α ∈ [ 0 , π 2 ) we have 1 6 | e − z − ( 1 + z / n ) n | ≤ ∀ n ≥ 1 , sup n cos 2 α z ∈ S 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 r n on S α , the relation f ( A ) = lim n →∞ r n ( A ) defines an operator f ( A ) ∈ L ( H , H ) and it follows that || f ( A ) || H → H ≤ C α sup z ∈ S α | f ( z ) | Lemma For all α ∈ [ 0 , π 2 ) we have 1 6 | e − z − ( 1 + z / n ) n | ≤ ∀ n ≥ 1 , sup n cos 2 α z ∈ S α Corollary Let α and β satisfy 0 ≤ α < α + β < π 2 . Then ∀ t ∈ S β 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 π Let α ∈ [ 0 , ) . The family of operators E ( t ) , t ≥ 0 satisfies the following properties 2 ∀ t , s ≥ 0 , E ( t + s ) = E ( t ) E ( s ) ∀ t ≥ 0 , || E ( t ) || H → H ≤ 1 ∀ u 0 ∈ H , the map t �→ E ( t ) u 0 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
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