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  • 1. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Immunology and van Kampen’s large N expansion Magic 042 – Lecture 9 (or lecture 15) Carmen Molina-Par´ ıs Department of Applied Mathematics, School of Mathematics, University of Leeds, UK 24th of November 2008
  • 2. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Outline of the talk More immunology Lymphocyte recirculation B cells and antibodies T cells T cells and MHC proteins T cell receptor (TCR) Effector cytotoxic T cells Effector helper T cells Helper T cells and lymphocyte activation Helper T cells and lymphocyte activation Large N expansion Large N expansion Hard niche case Hard niche case Symmetric clonotyes
  • 3. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Back to some more immunology
  • 4. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Lymphocyte recirculation through peripheral lymphoid organs I • Pathogens generally enter the body through an epithelial surface, usually through the skin, gut, or respiratory tract. • To induce an adaptive immune response, microbial antigens must travel from these entry points to a peripheral lymphoid organ, such as a lymph node or the spleen, the sites where lymphocytes are activated. • The route and destination depend on the site of entry. • Lymphatic vessels carry antigens that enter through the skin or respiratory tract to local lymph nodes. • Antigens that enter through the gut end up in gut-associated peripheral lymphoid organs such as Peyer’s patches. • The spleen filters out antigens that enter the blood.
  • 5. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Lymphocyte recirculation through peripheral lymphoid organs II
  • 6. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Lymphocyte recirculation through peripheral lymphoid organs III • Dendritic cells will carry the antigen from the site of infection to the peripheral lymphoid organ, where they play a crucial part in activating T cells. • Only a tiny fraction of the total lymphocyte population can recognise a particular microbial antigen in a peripheral lymph organ (estimated to be between 10−4 and 10−5 of each class of lymphocyte). • How do these rare cells find an antigen presenting cell displaying their antigen? • Lymphocytes continuously circulate between one peripheral lymphoid organ and another via the lymph and blood. • The continuous recirculation between the blood and lymph ends only if a lymphocyte is activated by its specific antigen in a peripheral lymphoid organ. • The lymphocyte remains in the peripheral lymphoid organ, where it proliferates and differentiates into either effector cells or memory cells. • Many of the effector T cells leave the lymphoid organ via the lymph and migrate through the blood to the site of infection, whereas others stay in the lymphoid organ and help activate B cells or other T cells there.
  • 7. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Lymphocyte recirculation through peripheral lymphoid organs IV • Some effector B cells (plasma cells) remain in the peripheral lymphoid organ and secrete antibodies into the blood for days until they die. • Others migrate to the bone marrow, where they secrete antibodies into the blood for months or years. • The memory T and B cells produced join the recirculating pool of lymphocytes. • Lymphocyte recirculation depends on specific interactions between the lymphocyte cell surface and the surface of the endothelial cells lining the blood vessels in the peripheral lymphoid organs. • Lymphocytes adhere and then migrate out of the bloodstream into the nodes. • The lymphocytes initially adhere to the endothelial cells via homing receptors that bind to specific ligands (often called counter-receptors) on the endothelial cell surface. • Lymphocyte migration into lymph nodes depends on a homing receptor protein called L-selectin.
  • 8. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Lymphocyte recirculation through peripheral lymphoid organs V • The lymphocytes adhere weakly to the endothelial cells and roll slowly along their surface. • The rolling continues until another, much stronger adhesion system is called into play by the chemoattractant proteins (chemokines) secreted by endothelial cells. • This strong adhesion is mediated by members of the integrin family of cell adhesion molecules, which become activated on the lymphocyte surface. • The lymphocytes stop rolling and crawl out of the blood vessel into the lymph node.
  • 9. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Lymphocyte recirculation through peripheral lymphoid organs VI • The T and B cells initially enter the same region of a lymph node, but then different chemokines guide them to separate regions of the node. • Unless they encounter their antigen, both T and B cells soon leave the lymph node via efferent lymphatic vessels. • If they encounter their antigen, however, they are stimulated to display adhesion receptors that trap the cells in the node. • The cells accumulate at the junction between the T cell and B cell areas, where the rare specific T and B cells can interact, leading to their proliferation and differentiation
  • 10. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cells and antibodies I • Antibodies defend us against infection by binding to viruses and microbial toxins, thereby inactivating them. • When antibodies bind to invading pathogens, they also recruit some of the components of the innate immune system.
  • 11. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cells and antibodies II • Synthesised exclusively by B cells, antibodies are produced in billions of forms, each with a different amino acid sequence. • Collectively called immunoglobulins (abbreviated as Ig), they are among the most abundant protein components in the blood. • Mammals make five classes of antibodies, each of which mediates a characteristic biological response following antigen binding. • In this section, we discuss the structure and function of antibodies and how they interact with antigen. • All antibody molecules made by an individual B cell have the same antigen-binding site.
  • 12. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cells and antibodies III • The first antibodies made by a newly formed B cell are not secreted but are instead inserted into the plasma membrane, where they serve as receptors for antigen. • Each B cell has approximately 105 such receptors in its plasma membrane. • Each B cell clone produces a single species of antibody, with a unique antigen-binding site. • When an antigen (with the aid of a helper T cell) activates a naive or a memory B cell, that B cell proliferates and differentiates into an antibody-secreting effector cell. • Such effector cells make and secrete large amounts of soluble (rather than membrane-bound) antibody, which has the same unique antigen-binding site as the cell-surface antibody that served earlier as the antigen receptor.
  • 13. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cells and antibodies IV
  • 14. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cells and antibodies V • Effector B cells can begin secreting antibody while they are still small lymphocytes, but the end stage of their maturation pathway is a large plasma cell. • Plasma cells continuously secrete antibodies at the astonishing rate of about 5000 molecules per second. • Although most plasma cells die after several days, some survive in the bone marrow for months or years and continue to secrete antibodies into the blood, helping to provide long-term protection against the pathogen that stimulated their production.
  • 15. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case The antibody molecule I • The simplest antibodies are Y-shaped molecules with two identical antigen-binding sites, one at the tip of each arm of the Y. • The basic structural unit of an antibody molecule consists of four polypeptide chains, two identical light (L) chains (each containing about 220 amino acids) and two identical heavy (H) chains (each usually containing about 440 amino acids). • The molecule is composed of two identical halves, each with the same antigen-binding site. • Both light and heavy chains usually cooperate to form the antigen-binding surface.
  • 16. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case The antibody molecule II
  • 17. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cell development
  • 18. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case T cells versus B cells I • Like antibody responses, T cell-mediated immune responses are exquisitely antigen-specific, and they are at least as important as antibodies in defending vertebrates against infection. • Most adaptive immune responses, including most antibody responses, require helper T cells for their initiation. • Unlike B cells, T cells can help eliminate pathogens that would otherwise be invisible inside host cells. • T cell responses differ from B cell responses in at least two crucial ways. • First, T cells are activated by foreign antigen to proliferate and differentiate into effector cells only when the antigen is displayed on the surface of antigen-presenting cells, usually dendritic cells in peripheral lymphoid organs. • T cells require antigen-presenting cells for activation because the form of antigen they recognise is different from that recognized by B cells.
  • 19. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case T cells versus B cells II • Whereas B cells recognise intact protein antigens, for example, T cells recognise fragments of protein antigens that have been partly degraded inside the antigen-presenting cell. • Special proteins, called MHC proteins, bind to the peptide fragments and carry them to the surface of the antigen-presenting cell, where T cells can recognise them. • The second difference is that, once activated, effector T cells act only at short range, either within a secondary lymphoid organ or after they have migrated into a site of infection. • Effector B cells, by contrast, secrete antibodies that can act far away. • Effector T cells interact directly with another host cell in the body, which they either kill (as in the case of an infected host cell, for example) or signal in some way (as in the case of a B cell or macrophage, for example). • We shall refer to such host cells as target cells.
  • 20. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case T cells versus B cells III • Target cells must display an antigen bound to an MHC protein on their surface for a T cell to recognise them, they are also antigen-presenting cells. • There are three main classes of T cells: cytotoxic T cells, helper T cells, and regulatory (suppressor) T cells. • Effector cytotoxic T cells directly kill cells that are infected with a virus or some other intracellular pathogen. • Effector helper T cells help stimulate the responses of other cells: mainly macrophages, dendritic cells, B cells, and cytotoxic T cells. • Effector regulatory T cells suppress the activity of other cells, especially of self-reactive effector T cells.
  • 21. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case T cell receptor (TCR) I • T cell responses depend on direct contact with an antigen-presenting cell or a target cell. • T cell receptors (TCRs), unlike antibodies made by B cells, exist only in membrane-bound form and are not secreted. • For this reason, TCRs were difficult to isolate, and it was not until the 1980s that researchers identified their molecular structure.
  • 22. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case T cell receptor (TCR) II • TCRs resemble antibodies. • The three-dimensional structure of the extracellular part of a TCR has been determined by x-ray diffraction, and it looks very much like one arm of a Y-shaped antibody molecule. • Various co-receptors and cellcell adhesion proteins greatly strengthen the binding of a T cell to an antigen-presenting cell or a target cell.
  • 23. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen presentation by dendritic cells I • Naive cytotoxic or helper T cells must be activated to proliferate and differentiate into effector cells before they can kill or help their target cells, respectively. • This activation occurs in peripheral lymphoid organs on the surface of activated dendritic cells that display foreign antigen complexed with MHC proteins on their surface, along with co-stimulatory proteins. • Memory T cells can be activated by other types of antigen-presenting cells, including macrophages and B cells, as well as by dendritic cells. • Dendritic cells interact with T cells to present antigens that either activate or suppress the T cells. • Dendritic cells are located in tissues throughout the body, including the central and peripheral lymphoid organs. • Wherever they encounter invading microbes, they endocytose the pathogens or their products. • If the encounter is not in a lymphoid organ, the dendritic cells carry the foreign antigens via the lymph to local lymph nodes or gut-associated lymphoid organs.
  • 24. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen presentation by dendritic cells II
  • 25. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen presentation by dendritic cells III • The encounter with a pathogen activates pattern recognition receptors of the dendritic cell, which is thereby induced to mature from an antigen-capturing cell to an activated antigen-presenting cell that can activate T cells. • Dendritic cells have to be activated in order to activate naive T cells, and they can also be activated by tissue injury or by effector helper T cells. • Tissue injury is thought to activate dendritic cells by the release of heat shock proteins and uric acid crystals when cells die by necrosis rather than by apoptosis. • Activated dendritic cells display three types of protein molecules on their surface that have a role in activating a T cell to become an effector cell or a memory cell
  • 26. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen presentation by dendritic cells IV • (1) MHC proteins, which present foreign antigen to the TCR, • (2) co-stimulatory proteins, which bind to complementary receptors on the T cell surface, and • (3) cell-cell adhesion molecules, which enable a T cell to bind to the antigen-presenting cell for long enough to become activated, which is usually hours. • Activated dendritic cells secrete a variety of cytokines that can influence the type of effector helper T cell that develops, as well as where the T cell migrates after it has been stimulated. • Non-activated dendritic cells also have important roles. • They help induce self-reactive T cells to become tolerant, both in the thymus and in other organs. • Such dendritic cells present self-antigens in the absence of the co-stimulatory molecules required to activate naive T cells.
  • 27. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen presentation by dendritic cells V
  • 28. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Effector cytotoxic T cells I • Cytotoxic T cells protect vertebrates against intracellular pathogens such as viruses and some bacteria and parasites that multiply in the host-cell cytoplasm, where they are sheltered from antibody-mediated attack. • Cytotoxic T cells do this by killing the infected cell before the microbes can proliferate and escape from the infected cell to infect neighboring cells. • Once a cytotoxic T cell has been activated by an infected antigen-presenting cell to become an effector cell, it can kill any target cell harboring the same pathogen. • Using its TCR, the effector cytotoxic T cell first recognises a microbial antigen bound to an MHC protein on the surface of an infected target cell. • This causes the T cell to reorganize its cytoskeleton and focus its killing apparatus on the target. • Focus is achieved when the TCRs actively aggregate, along with various co-receptors, adhesion molecules, and signaling proteins at the T cell/target cell interface, forming an immunological synapse.
  • 29. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Effector cytotoxic T cells II • A similar synapse forms when an effector helper T cell interacts with its target cell. • In this way, effector T cells avoid delivering their signals to neighboring cells. • Once bound to its target cell, an effector cytotoxic T cell can employ one of two strategies to kill the target, both of which operate by inducing the target cell to kill itself by undergoing apoptosis. • In killing an infected target cell, the cytotoxic T cell usually releases a pore-forming protein called perforin. • In the second killing strategy, the cytotoxic T cell activates a death-inducing cascade in the target cell less directly. • A homotrimeric protein on the cytotoxic T cell surface, called Fas ligand, binds to transmembrane receptor proteins on the target cell called Fas.
  • 30. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Effector cytotoxic T cells III
  • 31. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Effector helper T cells I • In contrast to cytotoxic T cells, helper T cells are crucial for defense against both extracellular and intracellular pathogens. • They help stimulate B cells to make antibodies that help inactivate or eliminate extracellular pathogens and their toxic products. • They also activate macrophages to destroy any intracellular pathogens multiplying within the macrophage’s phagosomes, and they help activate cytotoxic T cells to kill infected target cells. • They can also stimulate a dendritic cell to maintain it in an activated state. • Once an antigen-presenting cell activates a helper T cell to become an effector cell, the helper cell can then help activate other cells. • It does this both by secreting a variety of co-stimulatory cytokines and by displaying co-stimulatory proteins on its surface.
  • 32. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Effector helper T cells II • When activated by its binding to an antigen on a dendritic cell, a naive helper T cell usually differentiates into either of two distinct types of effector helper cell, called TH1 and TH2. • TH1 cells are mainly involved in immunity to intracellular microbes and help activate macrophages, cytotoxic T cells, and B cells. • TH2 cells are mainly involved in immunity to extracellular pathogens, especially multicellular parasites, and they help activate B cells to make antibodies against the pathogen. • The nature of the invading pathogen and the types of innate immune responses it elicits largely determine which type of helper T cell develops.
  • 33. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Effector helper T cells III
  • 34. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case MHC protein
  • 35. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case CD4 and CD8 co-receptors
  • 36. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Processing of viral protein for presentation to CTLs
  • 37. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Processing of an extracellular protein
  • 38. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case T cell development
  • 39. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Positive and negative selection
  • 40. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Helper T cells and lymphocyte activation
  • 41. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Activation of T cells
  • 42. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Activation of helper T cells
  • 43. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case TH 1 cells activation
  • 44. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen binding to B cell receptors (BCRs) I
  • 45. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Antigen binding to B cell receptors (BCRs) II
  • 46. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cell activation I
  • 47. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case B cell activation II
  • 48. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Immune recognition molecules belong to the ancient Ig superfamily
  • 49. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Large N expansion I • Let us consider two cell populations and their bivariate competition process. • Let n ≡ number of cells of type 1 and n ≡ number of cells of type 2. • The master equation for the process is given by ∂P(n, n , t) = λn−1,n P(n−1, n , t)+λn,n −1 P(n, n −1, t)+µn+1,n P(n+1, n , t) ∂t + µn,n +1 P(n, n + 1, t) − (λnn + λnn + µnn + µnn )P(n, n , t) . (1) • We introduce the difference operators: En f (n, n ) = f (n + 1, n ), (2) En f (n, n ) = f (n, n + 1), (3) E−1 f (n, n ) n = f (n − 1, n ), (4) E−1 f n (n, n ) = f (n, n − 1). (5)
  • 50. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Large N expansion II • The master equation can be written as: ∂P(n, n , t) −1 = (En − 1)λnn P(n, n , t) + (E−1 − 1)λnn P(n, n , t) n ∂t + (En − 1)µnn P(n, n , t) + (En − 1)µnn P(n, n , t). (6) • Define a suitable transformation of variables that will allow us to go from the discrete set of variables (n, n ) to the continuous set of variables (ξ, ξ ). • The large N expansion allows us to go from a stochastic description to a deterministic one (plus fluctuations).
  • 51. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Expansion of the master equation I • We expect n to consist of a deterministic part plus fluctuations. • We introduce Ω – a parameter measuring the volume of the system, such that for large Ω the fluctuations are relatively small. • We define the following transformation (n −→ ξ) 1 n = Ωx(t) + Ω 2 ξ . 1 • We assume the fluctuations are of order Ω 2 . • We define the following transformation (n −→ ξ ) 1 n = Ωx (t) + Ω 2 ξ .
  • 52. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Expansion of the master equation II • Now, rather than a probability distribution P of n and n , we have a probability distribution Π of ξ and ξ . 1 1 Π(ξ, ξ , t) = P(Ωx + Ω 2 ξ, Ωx + Ω 2 ξ , t) . (7) • We can write ∂P ∂Π 1 dx ∂Π 1 dx ∂Π = − Ω2 − Ω2 , (8) ∂t ∂t dt ∂ξ dt ∂ξ where we have made use of the chain rule. • Notice that in order to derive the previous equation, we have made use of the fact that dξ 1 dx dξ 1 dx = −Ω 2 , and = −Ω 2 . dt dt dt dt
  • 53. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Expansion of the master equation III • The translation operators can be written as follows 1 ∂ 1 ∂2 En = 1 + Ω− 2 + Ω−1 2 + . . . , (9) ∂ξ 2 ∂ξ 1 ∂ 1 ∂2 E−1 n = 1 − Ω− 2 + Ω−1 2 + . . . , (10) ∂ξ 2 ∂ξ 2 1 ∂ 1 ∂ En = 1 + Ω− 2 + Ω−1 + ..., (11) ∂ξ 2 ∂ξ 2 1 ∂ 1 ∂2 E−1 n = 1 − Ω− 2 + Ω−1 + ..., (12) ∂ξ 2 ∂ξ 2 • We make use of these expansions in the master equation to write ∂P(n, n , t) ∂Π(ξ, ξ , t) 1 dx ∂Π(ξ, ξ , t) 1 dx ∂Π(ξ, ξ , t) = − Ω2 − Ω2 (13) ∂t ∂t dt ∂ξ dt ∂ξ = (E−1 − 1)λnn P(n, n , t) + (E−1 − 1)λnn P(n, n , t) (14) n n +(En − 1)µnn P(n, n , t) + (En − 1)µnn P(n, n , t) . (15)
  • 54. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Expansion of the master equation IV • We know the birth and death rates depend both on n and n . • We need to express the translation operators in terms of the new variables ξ, ξ . • We need to express the birth and death rates in terms of the new variables ξ, ξ . 1 • The previous equation is a power series in Ω 2 . 1 • Identify terms of order Ω 2 ⇒ equation for x(t) and x (t) (deterministic part). • Identify terms of order Ω0 ⇒ master equation for Π(ξ, ξ , t) (fluctuations).
  • 55. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Hard niche case I • In the case where νii 1, νi/i 1, νi i 1, νi /i 1, we have the following birth and death rates: ϕ λnn = (n + n − pn ), (16) n+n ϕ λnn = (n + n − p n), (17) n+n µnn = µn, (18) µnn = µn, (19) with ϕp = ϕ p . • We make use of these birth and death rates and the variable transformations (n −→ ξ) and (n −→ ξ ) to rewrite the master equation in terms of the new variables.
  • 56. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Hard niche case II • The master equation (15) becomes 1 dx ∂Π∂Π 1 dx ∂Π − Ω2 − Ω2 dt ∂ξ∂t dt ∂ξ 2 3 1 −1 ∂ 2 " # −1 ∂ Ωϕ˜ −1 (ξ + ξ )(x + (1 − p)x ) « „ = 4−Ω 2 + Ω + . . .5 x + (1 − p)x + Ω 2 ξ + (1 − p)ξ − Π ∂ξ 2 ∂ξ2 x +x x +x 2 3 1 −1 ∂ 2 " # −1 ∂ Ωϕ˜ −1 (ξ + ξ )(x + (1 − p )x) « „ + 4−Ω 2 + Ω + . . .5 x + (1 − p )x + Ω 2 ξ + (1 − p )ξ − Π ∂ξ 2 ∂ξ 2 x +x x +x 2 3 2 −1 ∂ 1 −1 ∂ 1 + 4Ω 2 + Ω + . . .5 µ(Ωx + Ω 2 ξ)Π ∂ξ 2 ∂ξ2 2 3 2 −1 ∂ 1 −1 ∂ 1 + 4Ω 2 + Ω + . . .5 µ (Ωx + Ω 2 ξ )Π . ∂ξ 2 ∂ξ 2 • We have defined ϕΩ = ϕ and ϕ Ω = ϕ in the previous equations. ˜ ˜
  • 57. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Hard niche case III 1 • Terms of order Ω 2 result in the following deterministic equations: dx ϕ ˜ = (x + x − px ) − µx, (20) dt x +x dx ϕ ˜ = (x + x − p x) − µ x . (21) dt x +x • Terms of order Ω0 give a multi-variate linear Fokker-Plank equation: " # ∂Π ϕ ˜ ϕ ˜ ∂ = − + (x + x − px ) + µ (ξΠ) ∂t x +x (x + x )2 ∂ξ " # ϕ ˜ ϕ˜ ∂ + − + (x + x − p x) + µ (ξ Π) x +x (x + x )2 ∂ξ " # ϕ ˜ ϕ ˜ ∂ + − (1 − p) + (x + x − px ) (ξ Π) x +x (x + x )2 ∂ξ " # ϕ ˜ ϕ˜ ∂ + − (1 − p ) + (x + x − p x) (ξΠ) x +x (x + x )2 ∂ξ " # 2 1 ϕ ˜ ∂ Π + (x + x − px ) + µx 2 x +x ∂ξ2 " # 2 1 ϕ ˜ ∂ Π + (x + x − p x) + µ x . (22) 2 x +x ∂ξ 2
  • 58. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Hard niche case IV • The solution to the previous master equation is Gaussian. • It is fully determined by its first and second moments. • Thus, we then have the distribution of the fluctuations about the 1 deterministic value, to order Ω− 2 . • This is referred to as the linear noise approximation. • We introduce the following parameters: x+x (1−p) x +x(1−p ) K1 = x+x , K2 = x+x , K3 = 1 (x + x (1 − p)), 2 K4 = 1 (x + x(1 − p )). 2
  • 59. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Hard niche case V • From Eq. (22) above we have: " # d ϕ ˜ ` ´ ϕ ˜ ξ = (1 − K1 ) − µ ξ + (1 − p) − K1 ξ , (23) dt x +x x +x " # d ϕ ˜ ` ´ ϕ ˜ ξ = (1 − K2 ) − µ ξ + (1 − p ) − K2 ξ , (24) dt x +x x +x " # d 2 ϕ ˜ 2 ` ´ ϕ ˜ ξ = 2 (1 − K1 ) − µ ξ + 2 (1 − p) − K1 ξξ dt x +x x +x 2ϕ˜ + K3 + µx, (25) x +x " # d 2 ϕ ˜ 2 ` ´ ϕ ˜ ξ = 2 (1 − K2 ) − µ ξ + 2 (1 − p ) − K2 ξξ dt x +x x +x 2ϕ ˜ + K4 + µ x , (26) x +x " # d ϕ ˜ ` ´ ϕ ˜ 2 ξξ = (1 − K1 ) − µ ξξ + (1 − p) − K1 ξ dt x +x x +x " # ϕ ˜ ` ´ ϕ ˜ 2 + (1 − K2 ) − µ ξξ + (1 − p ) − K2 ξ . (27) x +x x +x
  • 60. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes I • We now consider the symmetric case where ϕ = ϕ , p = p and µ = µ . • The deterministic equations (20) and (21) become: dx ϕ ˜ = (x + x − px ) − µx, (28) dt x +x dx ϕ ˜ = (x + x − px) − µx . (29) dt x +x • The stationary state of this system of equations is found to be ϕ(2 − p) ˜ xs = xs = . (30) 2µ • Linear stability analysis shows that this stationary solution is stable. • We now consider the fluctuations about this stationary state.
  • 61. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes II • Equations (23) and (24) (for the time evolution of the means of the fluctuations) become: » – d µp pµ ξ = −µ ξ − ξ , (31) dt 2(2 − p) 2(2 − p) » – d µp pµ ξ = −µ ξ − ξ . (32) dt 2(2 − p) 2(2 − p) • These equations have the following stationary solution ξ s = ξ s =0. • This stationary state is stable.
  • 62. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes III • With initial conditions ξ |(t = 0) = ξ 0 and ξ |(t = 0) = ξ 0, the above system can be solved to give ξ + ξ 0 0 − ξ ξ 0 0 pµ ( 2−p −µ)t ξ = e −µt + e , (33) 2 2 ξ 0+ ξ 0 ξ 0− ξ 0 pµ ( 2−p −µ)t ξ = e −µt − e . (34) 2 2 • It is clear that ξ → 0 as t → +∞ and ξ → 0 as t → +∞. • Note also that if at t = 0, ξ = ξ = 0, then ξ = ξ = 0 at all subsequent times. • Then in terms of the original variables, to the present order, ϕ(2 − p) n = n = . 2µ
  • 63. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes IV • Equations (25), (26) and (27) become: » – d 2 µp pµ ξ = 2 −µ ξ2 − ξξ + ϕ(2 − p), ˜ dt 2(2 − p) 2−p » – d µp pµ ξ2 = 2 −µ ξ2 − ξξ + ϕ(2 − p), ˜ dt 2(2 − p) 2−p » – d µp pµ pµ ξξ = 2 −µ ξξ − ξ2 − ξ2 . dt 2(2 − p) 2(2 − p) 2(2 − p) • These equations have the following stationary solutions: ϕ(2 − p)(3p − 4) ˜ ξ2 s = , (35) 8µ(p − 1) ϕ(2 − p)(3p − 4) ˜ ξ2 s = , (36) 8µ(p − 1) ϕp(2 − p) ˜ ξξ s = . (37) 8µ(p − 1)
  • 64. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes V • The stationary solution is stable. • Note that ξ 2 s = ξ2 S ≥ 0 and ξξ s ≤ 0. • In terms of the original variables, the stationary variance of the number of T cells of clonotypes i and i is given by (3p − 4) 2 2 σn = σn = Ω ξ 2 = n . (38) 4(p − 1) • The covariance is given by 2 p σnn = Ω ξξ = n ≤ 0, (39) 4(p − 1) so there is a negative correlation between n and n .
  • 65. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes VI • With initial conditions ξ 2 |(t = 0) = ξ 2 0 , ξ 2 |(t = 0) = ξ 0 , and ξξ |(t = 0) = ξξ 0, the system of differential equations (35)-(35) can be solved to give: ϕ(2 − p)(3p − 4) ˜ µ(3p−4) 4µ(p−1) + c1 e −2µt + c2 e 2−p + c3 e 2−p , 40) t t ξ2 = ( 8µ(p − 1) ϕ(2 − p)(3p − 4) ˜ µ(3p−4) 4µ(p−1) + c1 e −2µt − c2 e 2−p + c3 e 2−p , 41) t t ξ2 = ( 8µ(p − 1) ϕ(2 − p)p ˜ 4µ(p−1) t ξξ = + c1 e 2µt − c3 e 2−p . (42) 8µ(p − 1)
  • 66. More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case Symmetric clonotyes VII • We have introduced the following parameters 3 1` 2 ´ 2ϕ(2 − p) ˜ c1 = ξξ 0 + ξ 0− ξ2 0 − , (43) 2 4 8µ(p − 1) 1` 2 2 ´ c2 = ξ 0 − ξ 0 , (44) 2 2 1 1` 2 ϕ(2 − p) ˜ − ξ2 ´ c3 = ξξ 0 + ξ 0 0 − . (45) 2 4 8µ(p − 1) • Note that as t → +∞, ξ 2 → ξ 2 s , ξ 2 → ξ 2 s , and ξξ → ξξ s.

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