Roles of repertoire diversity in robustness of humoral immune response
Alexander Mozeika, Franca Fraternali, Deborah Dunn-Walters, Anthony C. C. Coolen
DD R A F T Roles of repertoire diversity in robustness ofhumoral immune response
Alexander Mozeika , Franca Fraternali , Deborah Dunn-Walters , and Anthony C. C. Coolen Institute for Mathematical and Molecular Biomedicine, King’s College London, Hodgkin Building, London, UK; Randall Division of Cell and Molecular Biophysics, School ofBasic and Medical Biosciences, King’s College London, London, UK; Faculty of Health and Medical Sciences, School of Biosciences and Medicine, University of Surrey,Guildford, Surrey, UK; Department of Mathematics, King’s College London, The Strand, London, UKThis manuscript was compiled on October 30, 2019
The adaptive immune system relies on diversity of its repertoire of receptors to protect the organism from a great variety of pathogens. Sincethe initial repertoire is the result of random gene rearrangement, binding of receptors is not limited to pathogen-associated antigens butalso includes self antigens. There is a fine balance between having a diverse repertoire, protecting from many different pathogens, andyet reducing its self-reactivity as far as possible to avoid damage to self. In the ageing immune system this balance is altered, manifestingin reduced specificity of response to pathogens or vaccination on a background of higher self-reactivity. To answer the question whetherage-related changes of repertoire in the diversity and self/non-self affinity balance of antibodies could explain the reduced efficacy of thehumoral response in older people, we construct a minimal mathematical model of the humoral immune response. The principle of leastdamage allows us, for a given repertoire of antibodies, to resolve a tension between the necessity to neutralise target antigens as quickly aspossible and the requirement to limit the damage to self antigens leading to an optimal dynamics of immune response. The model predictsslowing down of immune response for repertoires with reduced diversity and increased self-reactivity. adaptive immune response | immune repertoire | repertoire diversity | repertoire self-reactivity T he adaptive immune system relies on an extremely diverse repertoire of receptors that can recognise target moleculesto protect us from pathogens. Each cell has a unique specificity, encoded by the T cell receptor on T cells, or the B cellreceptor on B cells. In the case of B cells, the B cell receptor is also known as surface immunoglobulin, and this immunoglobulin(Ig) can be secreted as antibody once the cell has developed into a plasma cell. Antibodies (Ab) are an important first line ofdefence, they can block the action of harmful target molecules and help to recruit additional elements of the immune system byacting as bridges between target molecules and effector cells. The targets of Ab are known as antigens (Ag).B cells are formed in the bone marrow, where they acquire a unique Ig via gene rearrangement, a process that can produceover 10 different genes by reassortment of less than 200 germline gene segments (1, 2). The highest diversity is seen inthe areas of the Ig gene where different gene segments are joined together, and these areas of the gene encode the partsof the Ab that bind to Ag, thus ensuring a large diversity in the Abs structural forms of possible binding interactions (3).Since gene rearrangement is essentially random, the potential binding interactions of the initial repertoire are not limited topathogen-associated target Ag, they can include self-Ag also. Immunological tolerance is a negative selection process wherebyB cells having Ig with strong binding to self are deleted from the repertoire so that they cannot develop into plasma cellssecreting self-reactive Abs (4). There is a trade-off between having a large enough shape space to be prepared for many differentpathogen-associated Ags and yet reducing self-reactivity as far as possible to avoid self-damage (5). During activation of Bcells in an immune response, the B cells with specificity for target Ag are expanded (6). With the advent of high throughputsequencing methods, we can see that there are a broad range of antibodies that respond, even for simple antigens such as tetanustoxin (7). The affinity for target Ag can be increased in germinal centres of secondary lymphoid tissue where B cells undergocycles of somatic hypermutation of their Ig genes, followed by competitive selection for the best target Ag-binders (8, 9). Thus,the initial repertoire is altered by both positive and negative selection events, depending on binding to target and self Ags.Older people are more susceptible to infection, in particular to bacterial infections such as pneumonia or urinary tractinfections (2). In the ageing immune system, the balance of the immune system is altered, manifested in a reduced specifictarget Ab response to infection or vaccination on a background of a higher number of Abs showing evidence of self-reactivity (8).In this instance, the presence of self-reactive Abs does not usually indicate autoimmune disease pathology, rather we believe itmay reflect an increased presence of ‘polyspecific’ or ‘promiscuous’ antibodies which have binding affinities that are measurablefor several different targets. Since we know that T cell availability and function is also compromised with age (10), it is possiblethat the B cell repertoire is not receiving as much help to produce affinity-matured specific antibodies that can dominate theimmune response, relying instead on more T-independent responses. Increased use of IgG2 over IgG1 detected in the samplesof older patients supports this hypothesis (11). Analyses of older Ig gene repertoires indicate that selection events at differentstages of B cell development, both positive and negative, are less effective in the older immune system (2). Some Ig genecharacteristics that have been associated with polyspecificity are seen to be increased in the naïve B cell population of olderpeople (12). In addition, a reduction in the diversity of the B cell repertoire overall has also been seen in older people (13).Our question is whether age-related repertoire changes in diversity and target/self-Ag affinity balance could explain thereduced efficacy of the humoral response in older people. To this end we construct a minimal mathematical model of thehumoral immune response. The ingredients of this model are Abs, target Ag and self-Ag. Abs are binding the target Agand thus reduce the amount of free target Ag, i.e. Ag not bound by Abs. The amount of free target Ag plays a role of an‘energy’ in our construction, and we assume that the immune system tries to minimise this energy. We note that various energy October 30, 2019 | vol. XXX | no. XX | a r X i v : . [ q - b i o . CB ] O c t R A F T functions have been used in immune system modelling in the past, such as the ‘total affinity’ in somatic hypermutation of Bcells (14), or the ‘disagreement’ between the B and T cell signalling in lymphocyte ‘networks’ in more recent studies (15–17).Furthermore, we assume that we have many types of Abs, each specified by its affinity to the targets and to self Ag (18), whichconstitute the immune repertoire in our model. Immune repertoires were studied theoretically in e.g. (19, 20), and more recentlyin (21). The role of self-Ags in shaping the diversity of repertoires, important for reliable self/non-self discrimination (19), wasemphasised in (20). We assume that both the binding of Abs to self-Ag and the presence of free target Ag incurs damage ,hence the unconstrained use of Abs is not possible and the amount of free target Ag has to be reduced. To resolve these twoconflicting requirements we develop the principle of least damage which allows us to derive an optimal dynamics of the immuneresponse. While the resulting theoretical framework is very general, even its simplest analytically solvable version predicts the‘slowing down’ of the immune response for repertoires with reduced diversity and increased self-reactivity. Mechanics of Immune Response
A simple thought experiment.
To investigate the trade-off between antibody binding to a desired target, such as pathogen,versus a self-damaging target, we consider the case where there are many antibodies responding to a challenge, in the absence ofa single dominating high-affinity antibody. Our thought experiment assumes that we have a finite volume reservoir containinga finite amount of target antigen (Ag) and self-antigen (self-Ag) in some medium (see Figure 1). We also assume that weare given M different types of antibodies (Abs), labelled by the integers 1 to M , which can be released into the reservoir.The release of each Ab is controlled by a valve. We assume that the reservoir contents are well mixed. Abs released into thereservoir react with both types of Ag, resulting in the formation of Ag-Ab complexes; thus the amount of ‘free’ (i.e. unbound)Ag is reduced. The properties of Abs, such as how strongly they react with each Ag, etc., are assumed to be initially unknown.Two gauges attached to the reservoir measure the amounts of free target Ag and of self-Ag. The opening and closing of valves,and performing various measurements (such as of the amount of Abs delivered into the reservoir, the amount of free target Agand self-Ag in the reservoir) constitutes an ‘experiment’. Measurement protocol.
The experimental measurement is defined by a set of time points t , . . . , t k − , t k , . . . , t n togetherwith the flow rates r µ ( t ) , . . . , r µ ( t k − ) , r µ ( t k ) , . . . , r µ ( t n ) recorded at these times, for each Ab µ (see Figure 1). We labelantibody types by Greek indices. The total amount of Ab µ released into the reservoir up to the time t k is given by the sum b µ ( t k ) = P k‘ =1 r µ ( t ‘ )( t ‘ − t ‘ − ). If the flow rates r µ ( t ) are smooth functions of time, each amount approaches an integral b µ ( t k ) = R t k t r µ ( t )d t in the limit where the measurement times become arbitrarily close, t ‘ − t ‘ − →
0. The system in Figure 1is then fully described by the amounts of Abs b ( t ) = ( b ( t ) , . . . , b M ( t )), delivered into the reservoir up to time t , and the rates dd t b ( t ) = (cid:0) dd t b ( t ) , . . . , dd t b M ( t ) (cid:1) of delivery of Abs. The amount of free target Ag, measured by the left gauge in Figure 1, is afunction A T ( b ( t )) of the Abs b ( t ). The same is true for A S ( b ), the amount of free self-Ag, measured by the right gauge in theFigure 1. By construction, the total amount of free Ag in the experiment is a non-increasing function of time, i.e. dd t A T ≤ dd t A S ≤ Measurement of antibody affinity.
Let the amount of free target Ag at time t be A T ( b ( t )), and assume that at the nexttime-point t we release into the reservoir a small amount ∆ b µ of Ab µ , i.e. b µ ( t ) = b µ ( t ) + ∆ b µ and b ν ( t ) = b ν ( t ) for all ν = µ . The resulting change in the amount of free target Ag is given by ∆ A µT = A T ( b ( t )) − A T ( b ( t )) ≤ b µ → ∂A T /∂b µ )(d b µ / d t ) ≤
0. The same holds for the free self-Ag A S ( b ). Upon releasing a single Ab into the reservoirwe will generally observe different behaviours of the gauges, which can be used to classify this Ab. Ab µ is more ‘reactive’than Ab ν if ∆ A µT ≤ ∆ A νT , for ∆ b µ = ∆ b ν , i.e. if the same amount of Ab reduces more Ag upon releasing type µ insterad of ν . Similarly, Ab µ is more self-reactive than Ab ν when ∆ A µS ≤ ∆ A νS , and Ab µ is more reactive than self-reactive when∆ A µT ≤ ∆ A µS (and vice versa). For ∆ b µ → µ ismore reactive than self-reactive when ( ∂A T /∂b µ ) ≤ ( ∂A S /∂b µ ), etc. Significance Statement
The older immune system is less able to protect us from infection and more likely to malfunction, and inappropriate inflammationis involved in the aetiology of many diseases of old age. Since the world population is growing older, immune senescence is asignificant health risk. Previous studies, by us and others, show that the human antibody repertoire is less diverse and there are moreantibodies that recognise self-antigens in older people. We posed the scenario that an antibody can bind multiple different targets,both self and non-self, but with varying affinity, and asked how efficacy of the immune system might be affected by this balance andby the loss of diversity of antibodies at a population level. Our theoretical framework was developed from first principles. It predictsthat a reduced diversity and increased self-reactivity in the antibody pool will slow down immune responses to exogenous targets,thus providing an explanation for the reduced immune response to vaccines and infections in older people.
A.M., F.F., D.D.-W. and A.C.C.C. designed research, performed research and wrote the paper.The authors declare no conflict of interest. † To whom correspondence should be addressed. E-mail: [email protected]
A.M., F.F., D.D.-W. and A.C.C.C. designed research, performed research and wrote the paper.The authors declare no conflict of interest. † To whom correspondence should be addressed. E-mail: [email protected] et al. R A F T Fig. 1.
Immune Response: the Thought Experiment.
Top drawing : antibodies (Abs) are released into a reservoir which contains a mixture of target antigen, Ag (red triangles)and self-antigen, self-Ag (blue circles). They can form Ab-Ag complexes and thereby reduce the amount of free (i.e. unbound) Abs, target Ag and self-Ag. The latter twoamounts are measured, respectively, by the left and right ‘gauges’. The experiment is performed under constraints , such as finite duration and finite reservoir volume.
Middledrawing: the release of antibodies is controlled by the flow rate (vertical axis) at any given time (horizontal axis). The total amount of Ab released up to time t k (crosses) isincreasing with time. Bottom drawing: the amount of free target Ag (self-Ag) is decreasing with time. Each measurement is taken at the time-point s k with s k (cid:29) t k , to ensurethat the mixture in the reservoir is always in equilibrium.Alexander Mozeika et al. PNAS |
October 30, 2019 | vol. XXX | no. XX | R A F T The difference ∆ A µT is related to the affinity of Ab µ (22), which is usually defined as the ratio r µ = K + µ /K − µ offorward/backward rates of the chemical reaction Ag + Ab (cid:10) AgAb . In chemical equilibrium the latter can be computedexperimentally, via the relation r µ = [ AgAb ] / [ Ag ][ Ab ], upon measuring the amount [ Ag ] of free target Ag, the amount [ Ab ] offree Ab, and the amount [ AgAb ] of Ag-Ab complexes, in the absence of other antibodies or antigens. In our notation, theaffinity can be written as r µ = − ([ Ag ] − [ AgAb ]) − [ Ag ][ Ab ] − Ag ] = − ∆ A µT ∆ b µ A T ( ) , [1]evaluated at b = . Thus for ∆ b µ → r µ ( b ) = − (cid:16) ∂∂b µ log A T ( b ) (cid:17) b = . [2]For b = , expression [2] can be seen as a generalised affinity , measured by adding a small amount of Ab µ in to the mixture ofAgs and Abs. The affinity to self-Ag r Sµ ( b ) uses the same definition as [2], but with A S ( b ) instead of A T ( b ).In immunology one commonly thinks in terms of a repertoire of different antibodies, each reacting to target-Ag or to self-Ag,and of changing repertoires representing expansions of target-Ag antibodies in immune activation and deletion of self-Agantibodies in immune tolerance. However, single antibodies can bind to multiple different antigens, with varying affinity, andthese antigens could be either target-Ag or self-Ag. What we may have empirically determined to be a specific target-Agbinding antibody may in fact be a polyspecific antibody where the binding to self-Ag is so small as to be unnoticed. So weneed to consider polyspecific antibodies, with variable affinities for binding to multiple Ag. Using multiple antibody types to reduce free antigen.
We assume here for simplicity that we have one type of target Ag, whichwe seek to reduce using a repertoire of antibodies. The Ag has N A distinct regions which can be ‘recognised’ by Abs, the epitopes . The Abs, represented by the amounts b = ( b , . . . , b M ), are assumed to interact with free epitopes, i.e. those not boundby Abs. The amounts of the free epitopes are written as E = ( E , . . . , E N A ). Each E i ≡ E i ( b ) must be a non-decreasing functionof the amount of Abs, such that 0 ≤ E i ( b ) ≤ E i ( ). Furthermore, the ‘amount’ of free target antigen A T ( b ) ≡ A T ( E ( b )) ≥ b ≡ b ( t ) of Abs in the reservoir (as in biological processes), i.e. thatdd t b µ = f µ ( b ) [3]For the dynamics [3] to reduce target Ag, it is sufficient that the rate functions f µ ( b ) are positive,dd t A T = M X µ =1 ∂A T ∂b µ dd t b µ = − A T ( b ) M X µ =1 r µ ( b ) f µ ( b ) ≤ . [4]Clearly, since A T ( b ) ≥
0, the A T ( b ) is a Lyapunov function of [3]. The possible choices for the Ab delivery rate functions f µ ( b )are further restricted by physical constraints in the experiment, such as finite time, finite volume, finite amount of availableAbs, etc. Further complications occur if, in addition to target Ag, the reservoir also contains self Ag and, when we try toreduce free target Ag, only a finite amount of reduced self Ag (off-target damage) can be tolerated. It is natural to assumethat the amount of free self Ag must depend in a similar way on the amount of free epitopes E S ( b ) = (cid:0) E S ( b ) , . . . , E SN S ( b ) (cid:1) asthe target antigen, so A S ( b ) = A S (cid:0) E S ( b ) (cid:1) . Furthermore, one would expect that the Ab dynamics [3] is also a function ofself-epitopes, i.e. dd t b µ = f µ (cid:0) E ( b ) , E S ( b ) (cid:1) , [5]and that any biologically sensible choice f µ ( . . . ) must be an increasing function of E ( b ) and a decreasing function of E S ( b ). Antibody Dynamics
Principle of least damage.
Instead of guessing an equation for the Ab delivery rates f µ ( . . . ), we take a Darwinian approachand assume that an optimized mechanism will have evolved that reduces the target Ag as quickly as possible , to minimise the‘damage’ done, while minimising the harmful binding to self Ag in the process. The optimization problem can be solved usingmathematical tools from physics. To this end we consider all possible paths b ( t ), allowed by the setup in Figure 1. Any suchpath will obey d b µ / d t ≥ A T / d t ≤
0, i.e. each will minimize A T ( b ) (which we will call the ‘potential energy’). Thelatter is a property of the reservoir. We assume that the antibody delivery mechanism in Figure 1 has associated with it a‘kinetic energy’ T (d b / d t ), which reflects the likely involvement of further variables governed by first order differential equations(equivalently, that the equations for b µ , if autonomous, will be at least second order). The path which begins at b ( t ) at time t and ends in b ( t ) at time t > t , with A T ( b ( t )) ≥ A T ( b ( t )), can then be obtained (23) by minimising the action S (cid:16) b , dd t b (cid:17) = Z t t d t L (cid:16) b ( t ) , dd t b ( t ) (cid:17) , [6]where L (cid:0) b , dd t b (cid:1) = A T ( b ) − T ( dd t b ) is the Lagrangian (see Materials and Methods ).
0, i.e. each will minimize A T ( b ) (which we will call the ‘potential energy’). Thelatter is a property of the reservoir. We assume that the antibody delivery mechanism in Figure 1 has associated with it a‘kinetic energy’ T (d b / d t ), which reflects the likely involvement of further variables governed by first order differential equations(equivalently, that the equations for b µ , if autonomous, will be at least second order). The path which begins at b ( t ) at time t and ends in b ( t ) at time t > t , with A T ( b ( t )) ≥ A T ( b ( t )), can then be obtained (23) by minimising the action S (cid:16) b , dd t b (cid:17) = Z t t d t L (cid:16) b ( t ) , dd t b ( t ) (cid:17) , [6]where L (cid:0) b , dd t b (cid:1) = A T ( b ) − T ( dd t b ) is the Lagrangian (see Materials and Methods ). et al. R A F T Interpretation of the action.
The area under the curve of A T ( b ( t )) on any path b ( t ), given by the integral D A ( t − t ) = Z t t A T ( b ( t )) d t, [7]can be seen as a damage inflicted upon the organism during the time interval [ t , t ] by the presence of free target Ag. Theintuition is that during any small time interval the damage inflicted by Ag is equal to the amount of free Ag times the time itspends in the organism. Definition [7] assumes moreover that this damage is cumulative , i.e. exposure to a large amount of Agfor a short time or a to a small amount of Ag for a longe time are equivalent. We observe that 0 ≤ D A ≤ A T ( b ( t )) ( t − t ),which follows from the properties A T ( b ( t )) ≥ A T ( b ( t )) ≥ A T ( b ( t )). So the path minimising the action [6] is the pathwhich minimises the damage D A ( t − t ), but subject to the constraint on d b / d t enforced by the term R t t d t T (cid:0) dd t b ( t ) (cid:1) inthe action (24).Similar to [7], we can consider the integral D S ( t − t ) = Z t t d t A S ( b ( t )) , [8]where 0 ≤ D S ≤ A S ( b ( t )) ( t − t ). From this integral follows the ‘damage to self’, defined for each small time interval as theamount of free self Ag reduced by off-target action of the Abs times the duration of this reduction. Thus during the interval[ t , t ] this damage is A S ( b ( t )) ( t − t ) − D S ( t − t ). Determination of optimal antibody dynamics.
We minimise the action [6] subject to the constraint [8], i.e. we assume thatremoval of some amount of self Ag can be tolerated. This is equivalent (24) to minimisation of [6] with the Lagrangian L (cid:16) b , dd t b (cid:17) = A T ( b ) − T (cid:16) dd t b (cid:17) − γA S ( b ) , [9]where γ is a Lagrange parameter. The solution of the minimization is described by the Euler-Lagrange equation (see Materialsand Methods ): dd t ∂∂ (d b µ / d t ) T (cid:16) dd t b (cid:17) = − ∂∂b µ [ A T ( b ) − γA S ( b )] . [10]We note that the above second order differential equations that describe the optimal control of antibody release were derivedfrom general system level principles, with only minimal and plausible assumptions. Their solution will involve 2 M constants,fixed by the boundary conditions b ( t ) and b ( t ).The natural form for the kinetic energy is T (d b / d t ) = P Mµ =1 Λ µ (d b µ / d t ) , where Λ µ >
0. It corresponds to assumingthat at least one set of further (as yet unspecified) variables play a role in the Ab delivery process. Insertion into [10] gives usthe ‘Newtonian’ equation Λ µ d d t b µ = A T ( b ) r µ ( b ) − γA S ( b ) r Sµ ( b ) , [11]where we used the affinities [2] to express the partial derivatives in [10]. We note that the Λ µ , which reflect properties of theAb delivery mechanism, act to introduce ‘inertia’: large (small) Λ µ reduce (increases) the tendency to change d b µ / d t . Thetotal ‘force’ Λ µ (d b µ / d t ) in [11] is a sum of a target Ag dependent term A T ( b ) r µ ( b ) that increases the rate of Ab delivery,and a self Ag dependent term − γA S ( b ) r Sµ ( b ) which decreases Ab delivery (if γ > µ (d b µ / d t ) = 0, marking the balance of forces in [11], gives us, for A S ( b ) , r µ ( b ) >
0, the identity A T ( b ) γA S ( b ) = r Sµ ( b ) r µ ( b ) . [12]It follows that there exists a function α ( b ) such that r µ ( b ) = α ( b ) r Sµ ( b ) for all µ . Furthermore, for b = the latter gives usthe relation r µ = α r Sµ between affinities, where α = α ( ). Results
Free Ag reduced by large numbers of ‘weak’ antibodies.
To proceed with our model we need to determine the dependencies of A T and A S on the antibody amounts b = ( b , . . . , b M ). Here we consider M distinct univalent Abs I µ , labelled by µ = 1 , . . . , M ,each interacting with the univalent target Ag ( ) and self-Ag ( ◦ ), via the following chemical reactions ◦ + I µ K S + µ (cid:10) K S − µ ◦ I µ + I µ K + µ (cid:10) K − µ I µ [13] Alexander Mozeika et al.
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October 30, 2019 | vol. XXX | no. XX | R A F T In chemical equilibrium, given the initial concentrations A T ( ) of the target Ag and A S ( ) of the self-Ag, the concentrations A T ( b ) of free target Ag and A S ( b ) of self-Ag are obtained by solving the following recursive system of equations; see Supplementary Information ( SI ), Section 1A: A T = A T ( )1 + P Mµ =1 b µ r µ A T r µ + A S r Sµ [14] A S = A S ( )1 + P Mµ =1 b µ r Sµ A T r µ + A S r Sµ . [15]Each Ab is characterised by its affinities to the target Ag, r µ = K + µ /K − µ (the ratio of forward and backward rates), and self-Ag, r Sµ = K S + µ /K S − µ . These give rise to the affinity vectors r = ( r , . . . , r M ) and r S = ( r S , . . . , r SM ), which define the Ab repertoire .For multiple self-Ags the repertoire is a matrix of affinities (see SI , Sections 1A & 2A).In order to use [11] one would prefer an explicit expression for A T ( b ) and A S ( b ), but how to solve the non-linear recursion[14] analytically is not clear. However, if we assume that affinities scale as r µ ≡ r µ /M and r Sµ ≡ r Sµ /M , then in the regime M → ∞ of having a large number of individually weak Abs, we obtain the concentrations of free Ags in explicit form (see Materials and Methods ): A T ( b ) = A T ( )1 + B ( b ) , A S ( b ) = A S ( )1 + B S ( b ) [16]expressed as functions of the averages B T ( b ) = 1 M M X µ =1 r µ b µ , B S ( b ) = 1 M M X µ =1 r Sµ b µ . The averages B T ( b ) and B S ( b ) can be seen as total affinities to the target Ag and the self Ag. A similar object, where b µ wasthe number of B cells with affinity to Ag r µ /M , was postulated as an ‘energy’ function of somatic hypermutation in (14).We note that the result [16], although derived for univalent Abs and Ag, is also true for multivalent Abs (see SI , Section1B). Thus our model predicts that it is possible to reduce target antigen without requiring affinity-matured antibodies, such asthose produced in a T-dependent reaction, if a sufficient number of weaker binders are available. Furthermore, the frameworkoutlined here can easily incorporate multiple Ags, chemical species binding Ab-Ag complexes, phagocytes, etc. (see SI , Section1A) Reduced macroscopic description.
Let us consider the Euler-Lagrange equations [11] for the free and self-Ag. Via [16], andupon reverting from the right-hand side of [11] back to that of its predecessor [10], these now take the formΛ µ d d t b µ = A T ( )(1 + B T ) r µ M − γ A S ( )(1 + B S ) r Sµ M , [17]where B T ≡ B ( b ) and B S ≡ B S ( b ). If we assume that Λ µ scales as Λ µ = λ µ φ ( M ) /M , where φ ( M ) = o ( M ), we can derive for M → ∞ the following equations ( SI , Section 2A):d d t B T = A T | r | (1 + B T ) − γ A S ( r · r S )(1 + B S ) [18]d d t B S = A T ( r · r S )(1 + B T ) − γ A S | r S | (1 + B S ) , where in the above we used the dot product definition x · y = M − P Mµ =1 λ − µ x µ y µ , with the associated norm | x | = √ x · x .We assume that at time t = 0 all Ab amounts and production rates are zero, i.e. b µ = d b µ / d t = 0 for all µ , so the initialconditions for [18] are B T (0) = B S (0) = 0 and (d B T / d t )(0) = (d B S / d t )(0) = 0. Furthermore, the average Ab concentration˜ B ( t ) = M − P Mν =1 b ν ( t ) is governed by the equationd d t ˜ B = A T ( r · )(1 + B T ) − γ A S ( r S · )(1 + B S ) . [19]with the short-hand = (1 , . . . , µ either r µ > r Sµ >
0, but never both. This implies that r · r S = 0, and that hence [18] decouples into two independent equations:d d t B T = A T | r | (1 + B T ) , d d t B S = − γ A S | r S | (1 + B S ) [20]
0, but never both. This implies that r · r S = 0, and that hence [18] decouples into two independent equations:d d t B T = A T | r | (1 + B T ) , d d t B S = − γ A S | r S | (1 + B S ) [20] et al. R A F T The dynamics of B T is now conservative , with energy function E (cid:16) B T , d B T d t (cid:17) = 12 | r | (cid:16) d B T d t (cid:17) + A T B T , [21]where the terms (d B T / d t ) / | r | and A T / (1 + B T ) are, respectively, the ‘kinetic’ and ‘potential’ energies. The equation for B T describes the motion of a ‘particle’ of ‘mass’ 1 / | r | in in a potential field (23). Furthermore, solving the energy conservationequation E ( B T , d B T / d t ) = E ( B T (0) , (d B T / d t )(0)), for B (0) = (d B T / d t )(0) = 0, gives usdd t B T = r A T | r | B T (1 + B T ) . [22]The function p B T / (1 + B T ) ∈ [0 ,
1] is monotonic increasing and concave for B T ≥
0. Hence t B ( t ) is bounded from aboveby p A T | r | t and this bound is saturated as t → ∞ . Also, the (normalised) amount of target antigen A T ( b ( t )) /A T ( ) =(1 + B T ( t )) − is bounded from below by (1 + p A T | r | t ) − .In a similar manner we simplify the dynamics of B S , which is also conservative, describing the motion of a particle of mass | r S | − and potential energy − γA S / (1 + B S ). Here we find (cid:16) dd t B S (cid:17) = − γA S | r S | B S (1 + B S ) [23]Since γ > B S = 0, i.e. self-reactive Abs are not used.We have now seen that [20] can be mapped into equations of Classical Mechanics. The equation for B S describes theacceleration of a particle of mass | r S | − in a gravitational field with gravitational constant γ , created by a another particle ofmass A S and radius one (23). The equation for B T has a similar interpretation but with a repulsive potential. Ag removal is faster in a more diverse repertoire, and slower when the repertoire has higher self-reactivity.
We return to themore general case where r · r S >
0, so Abs may have the potential to bind both target Ag and self Ag. Further analytic resultscan be obtained in the equilibrium regime of [18], defined by d B T / d t = d B S / d t = 0. This can only occur when r µ = αr Sµ for all µ (see SI , Section 2B) , where α >
0. The inverse α − can be seen as a degree of self-reactivity . From [17] it follows that B T = αB S in this regime, and that [18] can be reduced to a single equation:d d t B S = A S | r S | (cid:20) αβ (1 + αB S ) − γ (1 + B S ) (cid:21) , [24]with β = A T /A S . It is easy to show, using the above equation and [19], that now d ˜ B/ d t = ( r S · ) | r S | − d B S / d t , and hencethe average concentration of Abs is given by ˜ B = ( r S · ) | r S | − B S . [25]The dynamics [24] is again conservative, now with energy E (cid:16) B S , d B S d t (cid:17) = 12 | r S | (cid:16) d B S d t (cid:17) + βA S αB S − γA S B S . [26]As before we can use energy conservation, following initial conditions B S (0) = (d B S / d t )(0) = 0, to derivedd t B S = r A S | r S | (cid:16) βαB S αB S − γB S B S (cid:17) . [27]From this follows the following upper bound, which is saturated as t → ∞ (see SI , Section 2B): B S ( t ) ≤ t/τ, [28]with the time constant τ = 1 / | r S | p A S ( β − γ ) . [29]As a consequence of [28], we find for the normalised target Ag A T ( b ( t )) A T ( ) = 11 + αB S ( t ) ≥
11 + αt/τ . [30]So τ /α is a lower bound for the half-life of free target Ag; to achieve A T ( b ( t )) /A T ( ) = , the required time t has to be atleast τ /α . The lower bound for the half-life of self-Ag, derived by a similar argument, is found to be τ . Furthermore, if wedefine w ( λ ) = M − P Mµ =1 λ − µ then | r S | = p w ( λ ) [ σ λ ( r S ) + m λ ( r S )] , [31] Alexander Mozeika et al.
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October 30, 2019 | vol. XXX | no. XX | R A F T where σ λ (cid:0) r S (cid:1) = | r S | /w ( λ ) − (( r S · ) /w ( λ )) and m λ ( r S ) = ( r S · ) /w ( λ ) are, respectively, variance and mean of the self-affinities r S (see SI , Section 2B). Thus τ is monotonically decreasing with the variance σ λ ( r S ) and the mean m λ ( r S ). Sincethe former can be seen as a measure of the repertoire’s ‘diversity’, having a more diverse repertoire facilitates a more rapidreduction of target Ag.We also solved the differential equation [24] numerically for different inverse self-reactivities α . The solutions are plottedin Supplementary Information , in Figures 5-8. Comparison of the upper bound [28] with the solutions of [24] in Figure9 allows us to summarise various regimes. We first define, using [7], the normalised damage per unit time δ A ( t − t ) = D A ( t − t ) /A T ( b ( t )) ( t − t ), where 0 ≤ δ A ≤
1, and, using [8], the normalised damage to self per unit time 1 − δ S ( t − t ) =1 − D S ( t − t ) /A S ( b ( t )) ( t − t ), where 0 ≤ δ S ≤ ≤ − δ S ≤
1. For the system [16], on the time interval [0 , t ], theabove definitions give us δ A ( t ) = 1 t Z t d t αB S ( t ) , δ S ( t ) = 1 t Z t d t B S ( t ) [32]Now since (1 + αB S ) − is a monotonic decreasing function of B S , the upper bound [28] gives us the lower bounds δ A ( t ) ≥ τα t log (cid:16) α tτ (cid:17) [33] δ S ( t ) ≥ τt log (cid:16) tτ (cid:17) . [34]The latter gives us the upper bound 1 − ( τ /t ) log(1 + t/τ ) ≥ − δ S ( t ) for the damage to self.The two bounds on damages are plotted in Figure 2 for different values of self-reactivity constant α . For a repertoirewith Abs binding α times stronger to the target Ag than to the self-Ag the immune response is ‘normal’ and ‘autoimmune’,respectively, when α > α <
1. The normal response is characterised by a large decrease of free target Ag and a smalldecrease in free self-Ag per unit of time. For the autoimmune response it is the opposite. Furthermore, the normal response is‘accelerated’ by a larger α and increased repertoire diversity, but, for the same repertoire diversity, the autoimmune response isslower. Discussion
In this work we have shown, using only minimal assumptions, that antibody repertoire diversity is important in the effectiveremoval of antigen, in multiple ways. Not just because the repertoire will then have more chance of containing a single dominantantibody that can react to the target-Ag, but also because for a more diverse repertoire the half life of target-Ag will be smaller.Hence any decrease in repertoire diversity, such as that observed in older age, or caused by a prior immune response, can havean adverse effect on the immune response to challenge. Furthermore, reduction in efficacy of central tolerance mechanisms suchas can occur in older age, will result in greater self-reactivity in the repertoire, and this too will hamper an efficient immuneresponse against target-Ag.The mathematical framework in the form developed here can for now only be used to model the immune response to a finiteamount of Ag, with a fixed repertoire of Abs. Adaptation of the affinities of Abs to target Ag via affinity maturation (22) isnot yet included. To model the latter on could modify the Lagrangian [9], and derive dynamic equations for affinities. Also thepresent restriction on the amount of Ag can be relaxed within the current framework, by introducing (partially stochastic) Agreproduction and death.
Materials and Methods
The Variational Problem.
We aim to find the path b ( t ) that minimises the action [6] on the time-interval [ t , t ] with the boundaries b ( t ) = b and b ( t ) = b . This path must solve the equation δ S = 0 for the difference δ S = S ( b + δ b , d b / d t + d δ b / d t ) − S ( b , d b / d t ),where b ( t ) + δ b ( t ) is any perturbed path with δ b ( t ) = δ b ( t ) = 0 (24). Using the differential operator ∇ b = ( ∂/∂b , . . . , ∂/∂b M ) thisdifference, up to the order O (cid:0) | δ b | (cid:1) , can be written in the form δ S = Z t t L (cid:16) b + δ b , d b d t + δ d b d t (cid:17) d t − Z t t L (cid:16) b , d b d t (cid:17) d t [35]= Z t t n δ b . ∇ b L (cid:16) b , d b d t (cid:17) + δ d b d t . ∇ d b / d t L (cid:16) b , d b d t (cid:17)o d t = h δ b . ∇ d b / d t L (cid:16) b , d b d t (cid:17)i t t + · · ·· · · + Z t t δ b . n ∇ b L (cid:16) b , d b d t (cid:17) − dd t ∇ d b / d t L (cid:16) b , d b d t (cid:17) o d t = Z t t δ b . n ∇ b L (cid:16) b , d b d t (cid:17) − dd t ∇ d b / d t L (cid:16) b , d b d t (cid:17)o d t
We aim to find the path b ( t ) that minimises the action [6] on the time-interval [ t , t ] with the boundaries b ( t ) = b and b ( t ) = b . This path must solve the equation δ S = 0 for the difference δ S = S ( b + δ b , d b / d t + d δ b / d t ) − S ( b , d b / d t ),where b ( t ) + δ b ( t ) is any perturbed path with δ b ( t ) = δ b ( t ) = 0 (24). Using the differential operator ∇ b = ( ∂/∂b , . . . , ∂/∂b M ) thisdifference, up to the order O (cid:0) | δ b | (cid:1) , can be written in the form δ S = Z t t L (cid:16) b + δ b , d b d t + δ d b d t (cid:17) d t − Z t t L (cid:16) b , d b d t (cid:17) d t [35]= Z t t n δ b . ∇ b L (cid:16) b , d b d t (cid:17) + δ d b d t . ∇ d b / d t L (cid:16) b , d b d t (cid:17)o d t = h δ b . ∇ d b / d t L (cid:16) b , d b d t (cid:17)i t t + · · ·· · · + Z t t δ b . n ∇ b L (cid:16) b , d b d t (cid:17) − dd t ∇ d b / d t L (cid:16) b , d b d t (cid:17) o d t = Z t t δ b . n ∇ b L (cid:16) b , d b d t (cid:17) − dd t ∇ d b / d t L (cid:16) b , d b d t (cid:17)o d t et al. R A F T Fig. 2.
The damage due to antigen δ A (lower bound), plotted as a function of the damage to self − δ S (upper bound) for the inverse self-reactivity α = { − , − , − } (top red curves with α increasing from top to bottom) , α = 1 (black line) and α = { , , } (bottom blue curves with α increasing from top to bottom). The direction of‘time’ t/τ ∈ [0 , ∞ ) , indicated by arrows, is always from left to right.Alexander Mozeika et al. PNAS |
October 30, 2019 | vol. XXX | no. XX | R A F T where we used integration by parts and the stated boundary conditions. Solving δ S = 0 for the part of δ S that is linear in δ b gives us theso-called Euler-Lagrange equation dd t ∇ (d b / d t ) L (cid:16) b , d b d t (cid:17) = ∇ b L (cid:16) b , d b d t (cid:17) [36]with boundary conditions b ( t ) = b and b ( t ) = b . Mean-Field Limit.
Here we explain briefly the derivation of [16] from [14]. Substituting r µ → r µ /M and r Sµ → r Sµ /M into [14] gives A T A T ( ) = 11 + M P Mµ =1 r µ b µ (1 + A T r µ /M + A S r Sµ /M ) − [37]hence, if A T ( ) = φ ( M ) A T and A S ( ) = φ ( M ) A S , where φ ( M ) = o ( M ), i.e. lim M →∞ φ ( M ) /M = 0, then for M → ∞ we will indeedfind the mean-field expressiom [16] since A T A T ( ) = 11 + M P Mµ =1 r µ b µ + O ( φ ( M ) /M ) [38] ACKNOWLEDGMENTS.
This work was supported by the Medical Research Council of the United Kingdom (grant MR/L01257X/1).A.M. would like to thank Adriano Barra, Alessia Annibale, Fabián Aguirre López, Attila Csikász-Nagy and Martí Aldea Malo for veryenlightening discussions.
Supplementary Information1. Chemical kinetics of antigen-antibody reactions
A. Univalent antibodies reacting with univalent antigens .
We consider M different univalent antibodies (Abs), represented by the symbolsI µ with µ ∈ { , . . . , M } , forming complexes with M A different univalent target antigens (Ags), v with v ∈ { , . . . , M A } , and M S self-Ags, ◦ u with u ∈ { , . . . , M S } . The Ag bound by Ab v I µ and ◦ u I µ will subsequently form complexes with ‘phagocytic’ species P (22). Theformation and dissociation of complexes is modelled by the four chemical reactions ◦ u + I µ K S + µu (cid:10) K S − µu ◦ u I µ ◦ u I µ + P K + (cid:10) K − ◦ u I µ P v + I µ K + µv (cid:10) K − µv v I µ v I µ + P K + (cid:10) K − v I µ P . [39]In chemical equilibrium (25) the concentrations of free self-Ag, target Ag, Ab and P (denoted, respectively, by the symbols [ ◦ u ], [ v ], [I µ ]and [P]) are related to the concentration of bound species ◦ u I µ , ◦ u I µ P, v I µ and v I µ P (denoted, respectively, by the symbols [ ◦ u I µ ], [ ◦ u I µ P], [ v I µ ]and [ v I µ P]) via the affinity parameters r Sµu = K S + µu /K S − µu , r µv = K + µv K − µv and r = K + /K − , i.e. the ratios of forward/backward rates ofreactions: r Sµu = [ ◦ u I µ ][ ◦ u ] [I µ ] r = [ ◦ u I µ P][ ◦ u I µ ] [P] r µv = [ v I µ ][ v ] [I µ ] r = [ v I µ P][ v I µ ] [P] [40]Upon denoting the initial concentrations of the species ◦ u , v , I µ and P by [ ◦ u ] , [ v ] , [I µ ] and [P] , we can use mass conservation towrite [ ◦ u ] = [ ◦ u ] + M X µ =1 [ ◦ u I µ ] + M X µ =1 [ ◦ u I µ P] [41][ v ] = [ v ] + M X µ =1 [ v I µ ] + M X µ =1 [ v I µ P] [42][I µ ] = [I µ ] + M S X u =1 [ ◦ u I µ ] + M A X v =1 [ v I µ ] + M S X u =1 [ ◦ u I µ P] + M A X v =1 [ v I µ P] [43][P] = [P] + M X µ =1 M S X u =1 [ ◦ u I µ P] + M X µ =1 M A X v =1 [ v I µ P] [44]By using [40] these expressions can be written in the alternative form
We consider M different univalent antibodies (Abs), represented by the symbolsI µ with µ ∈ { , . . . , M } , forming complexes with M A different univalent target antigens (Ags), v with v ∈ { , . . . , M A } , and M S self-Ags, ◦ u with u ∈ { , . . . , M S } . The Ag bound by Ab v I µ and ◦ u I µ will subsequently form complexes with ‘phagocytic’ species P (22). Theformation and dissociation of complexes is modelled by the four chemical reactions ◦ u + I µ K S + µu (cid:10) K S − µu ◦ u I µ ◦ u I µ + P K + (cid:10) K − ◦ u I µ P v + I µ K + µv (cid:10) K − µv v I µ v I µ + P K + (cid:10) K − v I µ P . [39]In chemical equilibrium (25) the concentrations of free self-Ag, target Ag, Ab and P (denoted, respectively, by the symbols [ ◦ u ], [ v ], [I µ ]and [P]) are related to the concentration of bound species ◦ u I µ , ◦ u I µ P, v I µ and v I µ P (denoted, respectively, by the symbols [ ◦ u I µ ], [ ◦ u I µ P], [ v I µ ]and [ v I µ P]) via the affinity parameters r Sµu = K S + µu /K S − µu , r µv = K + µv K − µv and r = K + /K − , i.e. the ratios of forward/backward rates ofreactions: r Sµu = [ ◦ u I µ ][ ◦ u ] [I µ ] r = [ ◦ u I µ P][ ◦ u I µ ] [P] r µv = [ v I µ ][ v ] [I µ ] r = [ v I µ P][ v I µ ] [P] [40]Upon denoting the initial concentrations of the species ◦ u , v , I µ and P by [ ◦ u ] , [ v ] , [I µ ] and [P] , we can use mass conservation towrite [ ◦ u ] = [ ◦ u ] + M X µ =1 [ ◦ u I µ ] + M X µ =1 [ ◦ u I µ P] [41][ v ] = [ v ] + M X µ =1 [ v I µ ] + M X µ =1 [ v I µ P] [42][I µ ] = [I µ ] + M S X u =1 [ ◦ u I µ ] + M A X v =1 [ v I µ ] + M S X u =1 [ ◦ u I µ P] + M A X v =1 [ v I µ P] [43][P] = [P] + M X µ =1 M S X u =1 [ ◦ u I µ P] + M X µ =1 M A X v =1 [ v I µ P] [44]By using [40] these expressions can be written in the alternative form et al. R A F T [ ◦ u ] = [ ◦ u ] (cid:16) r [P]) M X µ =1 r Sµu [I µ ] (cid:17) [45][ v ] = [ v ] (cid:16) r [P]) M X µ =1 r µv [I µ ] (cid:17) [I µ ] = [I µ ] (cid:16) r [P]) ( M S X u =1 r Sµu [ ◦ u ] + M A X v =1 r µv [ v ] ) (cid:17) [P] = [P] (cid:16) r M X µ =1 ( M S X u =1 r Sµu [ ◦ u ] + M A X v =1 r µv [ v ] ) [I µ ] (cid:17) . Finally, upon introducing the notation A Su and A Tv for the concentrations [ ◦ u ] of free self Ags and [ v ] of target Ags we obtain the followingsystem of recursive equations, which, given the initial concentrations A Su ( ) ≡ [ ◦ u ] and A Tv ( ) ≡ [ v ] , b µ ≡ [I µ ] and P ( ) ≡ [P] , canbe used to obtain the equilibrium concentrations of free self and target Ag: A Su = A Su ( )1 + (1 + r [P]) P Mµ =1 r Sµu [I µ ] , A Tv = A Tv ( )1 + (1 + r [P]) P Mµ =1 r µv [I µ ] [46][I µ ] = b µ r [P]) (cid:8)P M S u =1 r Sµu A Su + P M A v =1 r µv A Tv (cid:9) [P] = P ( )1 + r P Mµ =1 (cid:8)P M S u =1 r Sµu A Su + P M A v =1 r µv A Tv (cid:9) [I µ ]We assume that the individual antibody affinities are weak, i.e. r Sµu ≡ r Sµu /M and r µv ≡ r µv /M , and consider r [P] = rP ( )1 + rM P Mµ =1 (cid:8)P M S u =1 r Sµu A Su + P M A v =1 r µv A Tv (cid:9) [I µ ] [47]= rP ( )1 + rM P Mµ =1 (cid:8)P M S u =1 r Sµu ˜ A Su A Su ( ) + P M A v =1 r µv ˜ A Tv A Tv ( ) (cid:9) [I µ ] , where we have defined the normalised concentrations ˜ A Tv = A Tv /A Tv ( ) and ˜ A Su = A Su /A Su ( ), in the limit M → ∞ of a ‘large’ numberof Ab types. If M A and M S are finite and A Su ( ) , A Tv ( ) , P ( ) ∝ φ ( M ), where we allow for φ ( M ) → ∞ as M → ∞ , but such that φ ( M ) /M →
0, i.e. φ ( M ) ∈ o ( M ), then r [P] = rP rM P Mµ =1 (cid:8)P M S u =1 r Sµu ˜ A Su A Su + P M A v =1 r µv ˜ A Tv A Tv (cid:9) [I µ ] + φ ( M ) , [48]where P ( ) = φ ( M ) P , A Su ( ) = φ ( M ) A Su and A Tv ( ) = φ ( M ) A Tv . Thus r [P] = O ( M ) when r Sµu , r µv = O ( M − ), M A , M S = O ( M )and A Su ( ) , A Tv ( ) , P ( ) = o ( M ). Using the above result in our equation for [I µ ] gives[I µ ] = b µ r [P]) (cid:8)P M S u =1 r Sµu A Su + P M A v =1 r µv A Tv (cid:9) [49]= b µ r [P]) (cid:8)P M S u =1 r Sµu ˜ A Su A Su + P M A v =1 r µv ˜ A Tv A Tv (cid:9) φ ( M ) M = b µ (cid:18) − (1 + r [P]) ( M S X u =1 r Sµu ˜ A Su A Su + M A X v =1 r µv ˜ A Tv A Tv ) φ ( M ) M + O (cid:18) φ ( M ) M (cid:19) (cid:19) = b µ + O ( φ ( M ) /M ) [50]Inserting this into equation [48] leads us for b = to r [P] = rP rM P Mµ =1 (cid:8)P M S u =1 r Sµu ˜ A Su A Su + P M A v =1 r µv ˜ A Tv A Tv (cid:9) (cid:8) b µ + O (cid:0) φ ( M ) M (cid:1)(cid:9) + φ ( M ) [51]= P P M S u =1 B Su ( b ) ˜ A Su A Su + P M A v =1 B v ( b ) ˜ A Tv A Tv + rφ ( M ) + O (cid:0) φ ( M ) rM (cid:1) = P P M S u =1 B Su ( b ) ˜ A Su A Su + P M A v =1 B v ( b ) ˜ A Tv A Tv + O (cid:16) rφ ( M ) (cid:17) Here we have defined the following two macroscopic observables: B Tv ( b ) = 1 M M X µ =1 r µv b µ , B Su ( b ) = 1 M M X µ =1 r Sµu b µ . Alexander Mozeika et al.
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October 30, 2019 | vol. XXX | no. XX | R A F T Finally, for the normalised self-Ag ˜ A Su = A Su /A Su ( ) and the normalised target Ag ˜ A Tv = A Tv /A Tv ( ) we proceed in a similar way andobtain the equations ˜ A Su = 11 + (1 + r [P]) P Mµ =1 r Sµu [I µ ] [52]= 11 + (1 + r [P]) M P Mµ =1 r Sµu b µ + O (cid:0) φ ( M ) M (cid:1) = 11+ (cid:18) P P MS ˜ u =1 B S ˜ u ( b ) ˜ A S ˜ u A S ˜ u + P MAv =1 B Tv ( b ) ˜ A Tv A Tv (cid:19) B Su ( b ) + O (cid:0) rφ ( M ) (cid:1) [53]˜ A Tv = 11+ (cid:18) P P MSu =1 B Su ( b ) ˜ A Su A Su + P MA ˜ v =1 B T ˜ u ( b ) ˜ A T ˜ u A T ˜ u (cid:19) B Tv ( b ) + O (cid:0) rφ ( M ) (cid:1) which for M → ∞ is equivalent to the system A Su = A Su ( )1+ (cid:18) P ( ) P MS ˜ u =1 B S ˜ u ( b ) A S ˜ u + P MAv =1 B Tv ( b ) A Tv (cid:19) B Su ( b ) , A Tv = A Tv ( )1+ (cid:18) P ( ) P MSu =1 B Su ( b ) A Su + P MA ˜ v =1 B T ˜ v ( b ) A T ˜ v (cid:19) B Tv ( b ) [54]These expressions hold when b = . If b = we simply have A Su = A Su ( ) and A Tv = A Tv ( ). We note that the affinity parameterlimit r → ∞ and the repertoire size limit M → ∞ commute. The meaning of the first limit is that the forward rate of the reaction AbAg + P (cid:10) AbAgP in [39] is much larger than the backward rate, i.e. K + (cid:29) K − . This limit enables us to use the present equilibriumframework to describe also irreversible processes, such as Ag ‘removal’ reactions like AbAg + P * AbAgP (26).The equations in [54] are functions of the sum y = P M S u =1 B Su A Su + P M A v =1 B Tv A Tv , which satisfies the recursive equation y = M S X u =1 A Su ( ) B Su (cid:0) P ( ) y (cid:1) B Su + M A X v =1 A Tv ( ) B Tv (cid:0) P ( ) y (cid:1) B Tv [55]= y M S X u =1 A Su ( ) B Su (1 + B Su ) y + P ( ) B Su + y M A X v =1 A Tv ( ) B Tv (1 + B Tv ) y + P ( ) B Tv = y M S X u =1 A Su ( ) B Su Q ˜ u = u (cid:2)(cid:0) B S ˜ u (cid:1) y + P ( ) B S ˜ u (cid:3)Q ˜ u (cid:2)(cid:0) B S ˜ u (cid:1) y + P ( ) B S ˜ u (cid:3) + y M A X v =1 A Tv ( ) B Tv Q ˜ v = v (cid:2)(cid:0) B T ˜ v (cid:1) y + P ( ) B T ˜ v (cid:3)Q ˜ v (cid:2)(cid:0) B T ˜ v (cid:1) y + P ( ) B T ˜ v (cid:3) , where B Su ≡ B Su ( b ) and B Tv ≡ B Tv ( b ). The above identity follows directly from the definition of y and [54]. Thus y is the solution of thefollowing polynomial equation, of order M S + M A : M S Y u =1 (cid:2)(cid:0) B Su (cid:1) y + P ( ) B Su (cid:3) M A Y v =1 (cid:2)(cid:0) B Tv (cid:1) y + P ( ) B Tv (cid:3) = M S X u =1 A Su ( ) B Su Y ˜ u = u (cid:2)(cid:0) B S ˜ u (cid:1) y + P ( ) B S ˜ u (cid:3) + M A X v =1 A Tv ( ) B Tv Y ˜ v = v (cid:2)(cid:0) B T ˜ v (cid:1) y + P ( ) B T ˜ v (cid:3) [56]Let us assume that the relevant solution of [56] is given by the function Φ (cid:0) B T , B S (cid:1) , where B T = (cid:0) B T , . . . , B TM A (cid:1) and B S = (cid:0) B S , . . . , B SM S (cid:1) , so that the solution of the recursion [54] is given by A Su (cid:0) B T , B S (cid:1) = A Su ( )Φ (cid:0) B T , B S (cid:1) (1 + B Su ) Φ ( B T , B S ) + P ( ) B Su , A Tv (cid:0) B T , B S (cid:1) = A Tv ( )Φ (cid:0) B T , B S (cid:1) (1 + B Tv ) Φ ( B T , B S ) + P ( ) B Tv [57]and the concentrations of (total) free self-Ag and target Ag are A S ( b ) = M S X u =1 A Su (cid:0) B T , B S (cid:1) A T ( b ) = M A X v =1 A Tv (cid:0) B T , B S (cid:1) . [58]For P ( ) = 0, i.e. in the absence of binding of Ag-Ab complexes to phagocytes, the above expressions simplify significantly to A S ( b ) = M S X u =1 A Su ( )1 + B Su ( b ) A T ( b ) = M A X v =1 A Tv ( )1 + B Tv ( b ) , [59]so the concentration of free Ag decreases with increasing concentrations of Abs. In Figure 3 we plot the (normalised) free target Agconcentration A T /A T ( ) = 1 / (1 + B ( b )) against the average concentration of Abs B T ( b ) = M − P Mµ =1 r µ b µ . For P ( ) > M A + M S [56], this could be non-nontrivial. But at least
October 30, 2019 | vol. XXX | no. XX | R A F T Finally, for the normalised self-Ag ˜ A Su = A Su /A Su ( ) and the normalised target Ag ˜ A Tv = A Tv /A Tv ( ) we proceed in a similar way andobtain the equations ˜ A Su = 11 + (1 + r [P]) P Mµ =1 r Sµu [I µ ] [52]= 11 + (1 + r [P]) M P Mµ =1 r Sµu b µ + O (cid:0) φ ( M ) M (cid:1) = 11+ (cid:18) P P MS ˜ u =1 B S ˜ u ( b ) ˜ A S ˜ u A S ˜ u + P MAv =1 B Tv ( b ) ˜ A Tv A Tv (cid:19) B Su ( b ) + O (cid:0) rφ ( M ) (cid:1) [53]˜ A Tv = 11+ (cid:18) P P MSu =1 B Su ( b ) ˜ A Su A Su + P MA ˜ v =1 B T ˜ u ( b ) ˜ A T ˜ u A T ˜ u (cid:19) B Tv ( b ) + O (cid:0) rφ ( M ) (cid:1) which for M → ∞ is equivalent to the system A Su = A Su ( )1+ (cid:18) P ( ) P MS ˜ u =1 B S ˜ u ( b ) A S ˜ u + P MAv =1 B Tv ( b ) A Tv (cid:19) B Su ( b ) , A Tv = A Tv ( )1+ (cid:18) P ( ) P MSu =1 B Su ( b ) A Su + P MA ˜ v =1 B T ˜ v ( b ) A T ˜ v (cid:19) B Tv ( b ) [54]These expressions hold when b = . If b = we simply have A Su = A Su ( ) and A Tv = A Tv ( ). We note that the affinity parameterlimit r → ∞ and the repertoire size limit M → ∞ commute. The meaning of the first limit is that the forward rate of the reaction AbAg + P (cid:10) AbAgP in [39] is much larger than the backward rate, i.e. K + (cid:29) K − . This limit enables us to use the present equilibriumframework to describe also irreversible processes, such as Ag ‘removal’ reactions like AbAg + P * AbAgP (26).The equations in [54] are functions of the sum y = P M S u =1 B Su A Su + P M A v =1 B Tv A Tv , which satisfies the recursive equation y = M S X u =1 A Su ( ) B Su (cid:0) P ( ) y (cid:1) B Su + M A X v =1 A Tv ( ) B Tv (cid:0) P ( ) y (cid:1) B Tv [55]= y M S X u =1 A Su ( ) B Su (1 + B Su ) y + P ( ) B Su + y M A X v =1 A Tv ( ) B Tv (1 + B Tv ) y + P ( ) B Tv = y M S X u =1 A Su ( ) B Su Q ˜ u = u (cid:2)(cid:0) B S ˜ u (cid:1) y + P ( ) B S ˜ u (cid:3)Q ˜ u (cid:2)(cid:0) B S ˜ u (cid:1) y + P ( ) B S ˜ u (cid:3) + y M A X v =1 A Tv ( ) B Tv Q ˜ v = v (cid:2)(cid:0) B T ˜ v (cid:1) y + P ( ) B T ˜ v (cid:3)Q ˜ v (cid:2)(cid:0) B T ˜ v (cid:1) y + P ( ) B T ˜ v (cid:3) , where B Su ≡ B Su ( b ) and B Tv ≡ B Tv ( b ). The above identity follows directly from the definition of y and [54]. Thus y is the solution of thefollowing polynomial equation, of order M S + M A : M S Y u =1 (cid:2)(cid:0) B Su (cid:1) y + P ( ) B Su (cid:3) M A Y v =1 (cid:2)(cid:0) B Tv (cid:1) y + P ( ) B Tv (cid:3) = M S X u =1 A Su ( ) B Su Y ˜ u = u (cid:2)(cid:0) B S ˜ u (cid:1) y + P ( ) B S ˜ u (cid:3) + M A X v =1 A Tv ( ) B Tv Y ˜ v = v (cid:2)(cid:0) B T ˜ v (cid:1) y + P ( ) B T ˜ v (cid:3) [56]Let us assume that the relevant solution of [56] is given by the function Φ (cid:0) B T , B S (cid:1) , where B T = (cid:0) B T , . . . , B TM A (cid:1) and B S = (cid:0) B S , . . . , B SM S (cid:1) , so that the solution of the recursion [54] is given by A Su (cid:0) B T , B S (cid:1) = A Su ( )Φ (cid:0) B T , B S (cid:1) (1 + B Su ) Φ ( B T , B S ) + P ( ) B Su , A Tv (cid:0) B T , B S (cid:1) = A Tv ( )Φ (cid:0) B T , B S (cid:1) (1 + B Tv ) Φ ( B T , B S ) + P ( ) B Tv [57]and the concentrations of (total) free self-Ag and target Ag are A S ( b ) = M S X u =1 A Su (cid:0) B T , B S (cid:1) A T ( b ) = M A X v =1 A Tv (cid:0) B T , B S (cid:1) . [58]For P ( ) = 0, i.e. in the absence of binding of Ag-Ab complexes to phagocytes, the above expressions simplify significantly to A S ( b ) = M S X u =1 A Su ( )1 + B Su ( b ) A T ( b ) = M A X v =1 A Tv ( )1 + B Tv ( b ) , [59]so the concentration of free Ag decreases with increasing concentrations of Abs. In Figure 3 we plot the (normalised) free target Agconcentration A T /A T ( ) = 1 / (1 + B ( b )) against the average concentration of Abs B T ( b ) = M − P Mµ =1 r µ b µ . For P ( ) > M A + M S [56], this could be non-nontrivial. But at least et al. R A F T A T A T ( ) B T Figure 13: Normalised antigen, AA ( ) , as a function of the average B = M P Mµ =1 r µ b µ .The e↵ect of this perturbation is to increase (or decrease) amount of availableAg and hence it models spontaneous “birth”, ↵ ( t ) (1 , b ), or “death” of the Ag.Assuming that A is a A -homogeneous, where A R , function of epitopes gives us A t ( b ) = ↵ A ( t ) A ( E ( b ) , . . . , E N A ( b )) (127)for some real . Furthermore, the antibody dynamics (8) is also becomes per-turbed and is given by the equationd b µ d t = ↵ µ ( t ) f µ ( {E i ( b ) : i @µ } ) , (128)where we assumed that f µ is also a homogeneous function of epitopes. For µ > ↵ µ ( t ) is a monotonic increasing in ↵ ( t ), so it is clear thatthe increase in Ag, due to its division, leads to the increase in Abs.Let us assume that ↵ ( t ) = 1 + e x ( t ) where ⌧ d x d t = ⇢x + ⌘ ( t ) (129)where ⌘ ( t ) is a zero-average Gaussian noise with h ⌘ ( t ) ⌘ ( t ) i = ⌧ ( t t ). For t ! 1 the random variable x ( t ) is governed by the distribution P ( x ) = 1 p ⇡/⇢ e ⇢ x (130) This is mathematically convenient but what is the physical meaning of this assumption? Fig. 3.
Normalised free antigen concentration, A T /A T ( ) , as a function of the average of Ab concentrations B T ( b ) = M − P Mµ =1 r µ b µ . for M A + M S = 2 we can compute this function analytically. Here Φ (cid:0) B T , B S (cid:1) ≡ Φ( B T , B S ) is the solution of the quadratic equation0 = (1 + B S ) (1+ B ) y + n B T (1+ B S ) [ P ( ) − A T ( )] + B S (1+ B T ) [ P ( ) − A S ( )] o y + B S B T P ( ) [ P ( ) − A S ( ) − A T ( )] . [60]Its determinant D = (cid:16) B T (1+ B S ) [ P ( ) − A T ( )] + B S (1+ B T ) [ P ( ) − A S ( )] (cid:17) [61] − B S B T (1+ B S ) (1+ B T ) P ( ) [ P ( ) − A S ( ) − A T ( )]is positive when A T ( ) + A S ( ) ≥ P ( ), in which case the equation has two real solutions. Only one of them is positive:Φ( B, B S ) = B T [ A T ( ) − P ( )]2 (1 + B T ) + B S [ A S ( ) − P ( )]2 (1 + B S ) [62]+ (cid:26) (cid:16) B T [ A T ( ) − P ( )]2 (1 + B T ) + B S [ A S ( ) − P ( )]2 (1 + B S ) (cid:17) + B T B S P ( ) [ A T ( )+ A S ( ) − P ( )](1 + B T ) (1 + B S ) (cid:27) . B. Bivalent Antibodies reacting with univalent target Antigen and self-Antigen.
In this section we show that in the regime of ‘weak’ Abs, asconsidered in previous section, the amount of free Ag is not affected by the valency of Abs (22). To this end it is sufficient only to considerthe case of bivalent Abs interacting with univalent target Ag and self-Ag. In particular we consider M different bivalent Abs, representedby the symbols Y µ with µ ∈ { , . . . , M } , forming complexes with univalent target Ag, , and univalent self-Ag, ◦ . The formation ofcomplexes is modelled by the following chemical reactions: ◦ + Y µ K S + µ (cid:10) K S − µ ◦ Y µ ◦ + ◦ Y µ K S + µ (cid:10) K S − µ ◦ ◦ Y µ + Y µ K N + µ (cid:10) K N − µ Y µ [63] + Y µ K N + µ (cid:10) K N − µ Y µ ◦ + Y µ K SN + µ (cid:10) K SN − µ ◦ 4 Y µ + ◦ Y µ K SN + µ (cid:10) K SN − µ Y µ [64]In chemical equilibrium, the concentrations of free self-Ag, target Ag, and Ab, which will be denoted, respectively, by the symbols [ ◦ ], [ ]and [Y µ ], are related to the concentrations of bound species ◦ Y µ , ◦ ◦ Y µ , Y µ , Y µ and Y µ , which we denote, respectively, by the symbols [ ◦ Y µ ],[ ◦ ◦ Y µ ], [ Y µ ], [ Y µ ] and [ Y µ ], via the affinities r Sµ = K S + µ /K S − µ , r Nµ = K N + µ /K N − µ and r SNµ = K SN + µ /K SN − µ via r Sµ = [ ◦ Y µ ][ ◦ ] [Y µ ] = [ ◦ ◦ Y µ ][ ◦ ] [ ◦ Y µ ] r Nµ = [ Y µ ][ ] [Y µ ] = [ Y µ ][ ] [ Y µ ] r SNµ = [ Y µ ][ ][ ◦ Y µ ] = [ ◦ 4 Y µ ][ ◦ ][ Y µ ] . [65] Alexander Mozeika et al.
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October 30, 2019 | vol. XXX | no. XX | R A F T If the initial concentrations of species ◦ , and Y µ are, respectively, given by [ ◦ ] , [ ] and [Y µ ] then, because of the mass conservation,we have [ ◦ ] = [ ◦ ] + M X µ =1 [ ◦ Y µ ] + M X µ =1 [ ◦ 4 Y µ ] + M X µ =1 [ Y µ ] + 2 M X µ =1 [ ◦ ◦ Y µ ] [66][ ] = [ ] + M X µ =1 [ Y µ ] + M X µ =1 [ ◦ 4 Y µ ] + M X µ =1 [ Y µ ] + 2 M X µ =1 [ Y µ ][Y µ ] = [Y µ ] + [ ◦ Y µ ] + [ Y µ ] + [ ◦ 4 Y µ ] + [ Y µ ] + [ ◦ ◦ Y µ ] + [ Y µ ] . Using the equilibrium relations [65] in the above three lines now gives us[ ◦ ] = [ ◦ ] P Mµ =1 (cid:2) r Sµ + r SNµ [ ] (cid:0) r Sµ + r Nµ (cid:1) + 2 r Sµ [ ◦ ] (cid:3) [Y µ ] [67][ ] = [ ] P Mµ =1 (cid:2) r Nµ + r SNµ [ ◦ ] (cid:0) r Sµ + r Nµ (cid:1) + 2 r Nµ [ ] (cid:3) [Y µ ] , where [Y µ ] = [Y µ ] r Sµ [ ◦ ] + r Nµ [ ] + r SNµ [ ◦ ] [ ] (cid:8) r Sµ + r Nµ (cid:9) + r Sµ [ ◦ ] + r Nµ [ ] . [68]Finally, with the notation A S = [ ◦ ], A S ( ) = [ ◦ ] , A T = [ ], A T ( ) = [ ] and b µ = [Y µ ] , we obtain the recursive equations A S = A S ( )1 + P Mµ =1 (cid:2) r Sµ + r SNµ ( r Sµ + r Nµ ) A T +2 r Sµ A S (cid:3) b µ r Sµ A S + r Nµ A T + r SNµ { r Sµ + r Nµ } A S A T + r Sµ A S + r Nµ A T [69] A T = A T ( )1 + P Mµ =1 (cid:2) r Nµ + r SNµ ( r Sµ + r Nµ ) A S +2 r Nµ A T (cid:3) b µ r Sµ A S + r Nµ A T + r SNµ { r Sµ + r Nµ } A S A T + r Sµ A S + r Nµ A T . Now let us redefine r µ = r µ /M , r Sµ = r Sµ /M and r SNµ = r SNµ /M , and consider the relevant term in our expression for A S : (cid:2) r Sµ + r SNµ (cid:0) r Sµ + r Nµ (cid:1) A T + 2 r Sµ A S (cid:3) b µ r Sµ A S + r Nµ A T + r SNµ (cid:8) r Sµ + r Nµ (cid:9) A S A T + r Sµ A S + r Nµ A T [70]= r Sµ M b µ + (cid:2) r SNµ (cid:0) r Sµ + r Nµ (cid:1) A T + 2 r Sµ A S (cid:3) b µ M (cid:2) r Sµ A S + r Nµ A T (cid:3) M + (cid:2) r SNµ (cid:8) r Sµ + r Nµ (cid:9) A S A T + r Sµ A S + r Nµ A T (cid:3) M = r Sµ b µ M + O (cid:0) φ ( M ) /M (cid:1) Here we assumed that A S , A ∝ φ ( M ), where φ ( M ) = o ( M ). The same argument applies to the corresponding term in the equation for A T , giving us r µ b µ /M + O (cid:0) φ ( M ) /M (cid:1) and hence A S ( b ) = A S ( )1 + M P Mµ =1 r Sµ b µ A T ( b ) = A T ( )1 + M P Mµ =1 r Nµ b µ [71]for M → ∞ , so we recover the result [59] for univalent Abs interacting with two types of Ag. The above argument easily generalises toinclude multiple univalent Ags and binding of Ag-Ab complexes.
2. Analysis of Antibody Dynamics
In this section we study the Euler-Lagrange equationΛ µ d d t b µ = − ∂∂b µ [ A T ( b ) − γA S ( b )] , [72]where Λ µ ≥ γ ≥
0, with the ‘energy’ functions A T ( b ) and A S ( b ) derived in section A. A. Binding of univalent Antigens by univalent Antibodies in the presence of univalent self-Antigens.
Let us define the total potential ‘energy’ A γ (cid:0) B T , B S (cid:1) = A T (cid:0) B T , B S (cid:1) − γA S (cid:0) B T , B S (cid:1) , [73]where A T (cid:0) B T , B S (cid:1) ≡ A T ( b ) and A S (cid:0) B T , B S (cid:1) ≡ A S ( b ), with A T ( b ) and A S ( b ) as defined in [58], and we consider equation [72] forthis energy function: Λ µ d d t b µ = − ∂∂b µ A γ (cid:0) B T , B S (cid:1) = − M A X k =1 ∂A γ ∂B Tk ∂B Tk ∂b µ − M S X ‘ =1 ∂A γ ∂B S‘ ∂B S‘ ∂b µ = − M A X k =1 ∂A γ ∂B Tk r µk M − M S X ‘ =1 ∂A γ ∂B S‘ r Sµ‘ M
Let us define the total potential ‘energy’ A γ (cid:0) B T , B S (cid:1) = A T (cid:0) B T , B S (cid:1) − γA S (cid:0) B T , B S (cid:1) , [73]where A T (cid:0) B T , B S (cid:1) ≡ A T ( b ) and A S (cid:0) B T , B S (cid:1) ≡ A S ( b ), with A T ( b ) and A S ( b ) as defined in [58], and we consider equation [72] forthis energy function: Λ µ d d t b µ = − ∂∂b µ A γ (cid:0) B T , B S (cid:1) = − M A X k =1 ∂A γ ∂B Tk ∂B Tk ∂b µ − M S X ‘ =1 ∂A γ ∂B S‘ ∂B S‘ ∂b µ = − M A X k =1 ∂A γ ∂B Tk r µk M − M S X ‘ =1 ∂A γ ∂B S‘ r Sµ‘ M et al. R A F T b b µ b M Ar µ r Sµ A S Figure 6: Network of M di↵erent populations of univalent antibodies (smallblue circles) interacting with populations of the univalent antigen (red triangle)and self-antigen (blue circle). Population of the antibody µ is interacting withthe antigen and self-Ag with, respectively, the ‘strength’ r µ , i.e. a nity , and r Sµ . Population of antigens is interacting with all antibodies.Let us define the averages B ( t ) = M P M⌫ =1 r ⌫ b ⌫ ( t ) and B S ( t ) = M P M⌫ =1 r S⌫ b ⌫ then above equation gives the closed system of equations¨ B = A ( ) | r | (1 + B ) A S ( ) h r , r S i (1 + B S ) (70)¨ B S = A ( ) h r , r S i (1 + B ) A S ( ) | r S | (1 + B S ) , where in above we used the definition of inner product (58) to represent averagessuch as M P Mµ =1 r µ r Sµ / µ etc., with the set of initial conditions { ˙ B (0) , ˙ B S (0) , B (0) , B S (0) } .We note that the ’observable’ B ( t ) = M P M⌫ =1 r ⌫ b ⌫ ( t ) and B S ( t ) = M P M⌫ =1 r S⌫ b ⌫ ( t )can be computed from the a nities r µ and r Sµ of the Ab µ measured indepen-dently from the other Ab’s. Furthermore, the observable ˜ B ( t ) = M P M⌫ =1 b ⌫ ( t )is governed by the equation¨˜ B = A ( ) h r , i (1 + B ) A S ( ) h r S , i (1 + B S ) . (71)The simplest case to consider is when each Ab is either self-reactive or non-self-reactive, i.e. for Ab µ either the a nity to non-self r µ = 0 and self r Sµ > r µ > r Sµ = 0. The latter implies that h r , r S i = 0, in thesystem of equations (70), giving us the two equations¨ B = A ( ) | r | (1 + B ) (72)¨ B S = A S ( ) | r S | (1 + B S ) Fig. 4.
Network representation of M different populations of univalent Abs (small blue circles) interacting with populations of univalent target Ag (red triangle) and self-Ag (largeblue circle). Ab µ is interacting with the target Ag and self-Ag with, strengths (affinities) r µ and r Sµ , respectively. The Ags are interacting with all Abs. Assuming that Λ µ = λ µ φ ( M ) /M , where φ ( M ) = o ( M ), and using definition [52] above, allows us to derive the following equations for theset of macroscopic observables B T and B S :d d t B Tv = − M A X k =1 ( r v · r k ) ∂∂B Tk A γ (cid:0) B T , B S (cid:1) − M S X ‘ =1 ( r v · r S‘ ) ∂∂B S‘ A γ (cid:0) B T , B S (cid:1) [74]d d t B Su = − M A X k =1 ( r Su · r k ) ∂∂B Tk A γ (cid:0) B T , B S (cid:1) − M S X ‘ =1 ( r Su · r S‘ ) ∂∂B S‘ A γ (cid:0) B T , B S (cid:1) [75]with the short-hand x · y = M − P Mµ =1 λ − µ x µ y µ , with associated inner product norm | x | = √ x · x .In the special simplified case where each Ab µ interacts with only one type of Ag, we will have r v · r k = 0 if v = k , r v · r S‘ = 0, etc.,and the system of equations [74] simplifies to1 | r v | d d t B v = − ∂∂B v A γ (cid:0) B , B S (cid:1) | r Su | d d t B Su = − ∂∂B Su A γ (cid:0) B , B S (cid:1) . We note that the above simplified macroscopic dynamics is conservative (23), with the energy function E (cid:16) B T , dd t B T ; B S , dd t B S (cid:17) = M A X v =1 | r v | (cid:16) d B Tv d t (cid:17) + M S X u =1 | r Su | (cid:16) d B Su d t (cid:17) + A γ (cid:0) B T , B S (cid:1) [76]where the first two terms play the role of ‘kinetic’ energies, and the third term is the ‘potential’ energy. The factors 1 / | r v | and 1 / | r Su | can be seen as ‘masses’. So [76] describes the motion (23) of M A + M S ‘particles’, with distinct masses, in a potential field with potentialenergy [73].Let us now assume that the numbers of target and self Ags are equal, i.e. M A = M S , and that each Ab µ simultaneously interacts withtwo types of Ag, one target and one self (see Figure 4 for M A = M S = 1). Then the affinity vectors r v and r Su satisfy the orthogonality conditions r v · r k = 0 if k = v and r Su · r S‘ = 0 if ‘ = u , i.e. each row in the affinity matrices R T = ( r , . . . , r M A ) and R S = ( r S , . . . , r SM A )has exactly one positive component. Also r v · r S‘ = 0 if ‘ = u , so, up to a permutation of columns, the matrices R T and R S are the same.Our equations then simplify to d d t B Tv = − ∂∂B Tv A γ (cid:0) B T , B S (cid:1) | r v | − ∂∂B Su A γ (cid:0) B T , B S (cid:1) ( r v · r Su ) [77]d d t B Su = − ∂∂B Tv A γ (cid:0) B T , B S (cid:1) ( r Su · r v ) − ∂∂B Su A γ (cid:0) B T , B S (cid:1) | r Su | . [78]Assuming that the above system is in ‘mechanical’ equilibrium , d B Tv / d t = d B Su / d t = 0, leads us to the two equalities − ∂A γ (cid:0) B T , B S (cid:1) /∂B Tv ∂A γ ( B T , B S ) /∂B Su = ( r v · r Su ) | r v | − ∂A γ (cid:0) B T , B S (cid:1) /∂B Tv ∂A γ ( B T , B S ) /∂B Su = | r Su | ( r Su · r v ) [79]and hence ( r v · r Su ) = | r Su | | r v | . [80] Alexander Mozeika et al.
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October 30, 2019 | vol. XXX | no. XX | R A F T We note that this will be true if and only if r = α ( v, u ) r Su , for some α ( v, u ) >
0. Using this in [77] gives us the equationsd d t B v = − ∂∂B v A γ (cid:0) B , B S (cid:1) α ( v, u ) | r Su | − ∂∂B Su A γ (cid:0) B , B S (cid:1) α ( v, u ) | r Su | [81]d d t B Su = − ∂∂B v A γ (cid:0) B , B S (cid:1) α ( v, u ) | r Su | − ∂∂B Su A γ (cid:0) B , B S (cid:1) | r Su | We note that α ( v, u ) generates a mapping r v = α ( v, u ) r Su between the affinities r v and r Su . Without loss of generality, we can alwaysre-label the antibodies such that u = v , so that we only need α ( v, v ) ≡ α ( v ). Equation [81] can then be simplified to1 α ( v ) | r Sv | d d t B Tv = − ∂∂B Tv A γ (cid:0) B T , B S (cid:1) α ( v ) − ∂∂B Sv A γ (cid:0) B T , B S (cid:1) [82]1 | r Sv | d d t B Sv = − ∂∂B Tv A γ (cid:0) B T , B S (cid:1) α ( v ) − ∂∂B Sv A γ (cid:0) B T , B S (cid:1) . Furthermore, since now B Tv = α ( v ) B Sv the above reduces to the single equation1 | r Sv | d d t B Sv = − α ( v ) ∂∂B Tv A γ (cid:0) B T , B S (cid:1) − ∂∂B Sv A γ (cid:0) B T , B S (cid:1) , [83]where the partial derivatives are evaluated at B Tv = α ( v ) B Sv .The macroscopic dynamics [83] is conservative when P ( ) = 0. In this case the potential energy [73] is given by A γ (cid:0) B T , B S (cid:1) = M A X v =1 A Tv ( )1 + B Tv − γ M S X u =1 A Su ( )1 + B Su [84]and equation [83] reduces to 1 | r Sv | d d t B Sv = − ∂∂B Sv (cid:26) A Tv α ( v ) B Sv − γ A Sv B Sv (cid:27) , [85]so this dynamics is conservative, with the energy E v (cid:16) B Sv , dd t B Sv (cid:17) = 12 | r Sv | (cid:16) dd t B Sv (cid:17) + A Tv (1 + α ( v ) B Sv ) − γ A Sv (1 + B Sv ) , [86]describing the ‘motion’ a ‘particle’ of ‘mass’ 1 / | r Sv | in a potential field. If at time t = 0 we are given the initial position B Sv (0) andvelocity (d B Sv / d t )(0) of this particle, then for all t > E v (cid:16) B Sv , dd t B Sv (cid:17) = E v (cid:16) B Sv (0) , ( dd t B Sv )(0) (cid:17) . [87] B. Binding of univalent Antigen by univalent Antibodies in the presence of univalent self-Antigen.
The dynamics [72] with the energy function[84] can be solved in a full detail when M A = M S = 1 (see Figure 4). Here the Euler-Lagrange equation isΛ µ d d t b µ = − ∂∂b µ h A T ( )1 + B T ( b ) − γ A S ( )1 + B S ( b ) i [88]= A T ( )(1 + B T ( b )) r µ M − γ A S ( )(1 + B S ( b )) r Sµ M , where B T ( b ) = M − P Mν =1 r ν b ν ( t ) and B S ( b ) = M − P Mν =1 r Sν b ν . The latter two macroscopic observables are governed by theequations d d t B = A T | r | (1 + B T ) − γ A S ( r · r S )(1 + B S ) d d t B S = A T ( r · r S )(1 + B T ) − γ A S | r S | (1 + B S ) , [89]where B T ≡ B ( b ) and B S ≡ B S ( b ), with initial conditions { (d B T / d t )(0) , (d B S / d t )(0) , B T (0) , B S (0) } . So the above equations are aspecial case of [74,75]. Furthermore, the average concentration of Abs ˜ B ( b ) = M − P Mν =1 b ν is governed byd d t ˜ B = A T ( r · )(1 + B T ) − γ A S ( r S · )(1 + B S ) . [90]The simplest case is that where each Ab is either self-reactive or non-self-reactive (never both), i.e. for all µ either r µ = 0 and r Sµ > r µ > r Sµ = 0. This implies that ( r · r S ) = 0 in [89], giving us the two independent equationsd d t B T = A T | r | (1 + B T ) d d t B S = − γ A S | r S | (1 + B S ) . [91]We note that above is a special case of [76], so the dynamics of B T is conservative with the energy E (cid:16) B T , dd t B T (cid:17) = 12 | r | (cid:16) dd t B T (cid:17) + A T B T , [92]Since energy is conserved, one can then use the identity E ( B T , d B T / d t ) = E ( B T (0) , (d B T / d t )(0)) to obtain a simple equation for d B/ d t .For the initial conditions (d B T / d t )(0) = B T (0) = 0 this equation is given bydd t B T = r A T | r | B T B T . [93]
The dynamics [72] with the energy function[84] can be solved in a full detail when M A = M S = 1 (see Figure 4). Here the Euler-Lagrange equation isΛ µ d d t b µ = − ∂∂b µ h A T ( )1 + B T ( b ) − γ A S ( )1 + B S ( b ) i [88]= A T ( )(1 + B T ( b )) r µ M − γ A S ( )(1 + B S ( b )) r Sµ M , where B T ( b ) = M − P Mν =1 r ν b ν ( t ) and B S ( b ) = M − P Mν =1 r Sν b ν . The latter two macroscopic observables are governed by theequations d d t B = A T | r | (1 + B T ) − γ A S ( r · r S )(1 + B S ) d d t B S = A T ( r · r S )(1 + B T ) − γ A S | r S | (1 + B S ) , [89]where B T ≡ B ( b ) and B S ≡ B S ( b ), with initial conditions { (d B T / d t )(0) , (d B S / d t )(0) , B T (0) , B S (0) } . So the above equations are aspecial case of [74,75]. Furthermore, the average concentration of Abs ˜ B ( b ) = M − P Mν =1 b ν is governed byd d t ˜ B = A T ( r · )(1 + B T ) − γ A S ( r S · )(1 + B S ) . [90]The simplest case is that where each Ab is either self-reactive or non-self-reactive (never both), i.e. for all µ either r µ = 0 and r Sµ > r µ > r Sµ = 0. This implies that ( r · r S ) = 0 in [89], giving us the two independent equationsd d t B T = A T | r | (1 + B T ) d d t B S = − γ A S | r S | (1 + B S ) . [91]We note that above is a special case of [76], so the dynamics of B T is conservative with the energy E (cid:16) B T , dd t B T (cid:17) = 12 | r | (cid:16) dd t B T (cid:17) + A T B T , [92]Since energy is conserved, one can then use the identity E ( B T , d B T / d t ) = E ( B T (0) , (d B T / d t )(0)) to obtain a simple equation for d B/ d t .For the initial conditions (d B T / d t )(0) = B T (0) = 0 this equation is given bydd t B T = r A T | r | B T B T . [93] et al. R A F T The function p B/ (1+ B ) ∈ [0 ,
1] is monotonic increasing and concave for B ∈ [ 0 , ∞ ). Hence B T ( t ) is bounded from above by p A | r | t ,saturating this upper bound as t → ∞ . Furthermore, the (normalised) amount of antigen A T /A ( ) = 1 / (1+ B T ( t )) is bounded frombelow by 1 / (1+ p A | r | ) t . Also the dynamics of B S in [91] is conservative, with energy E (cid:16) B S , dd t B S (cid:17) = 12 | r S | (cid:16) dd t B S (cid:17) − γA S B S , [94]and using E ( B S , d B S / d t ) = E ( B S (0) , (d B S / d t )(0)), with initial conditions (d B S / d t )(0) = B S (0) = 0, gives us the equation (cid:16) dd t B S (cid:17) = − γA S | r S | B S B S [95]which for γ > B S = 0. Values γ < B T / d t = d B S / d t = 0. From these conditions we infer that( r · r S ) = r ( r S ) , hence r µ = α r Sµ for some α >
0. This, in return, via the definitions of B T and B S , implies B T = αB S and hence thesystem [89] reduces to a single equation: d d t B S = A S | r S | (cid:20) αβ (1 + αB S ) − γ (1 + B S ) (cid:21) , [96]where we defined β = A T /A S . Furthermore, for equation [90], governing the average concentration of antibodies ˜ B , we obtaind d t ˜ B = A S ( r S · ) (cid:20) αβ (1 + αB S ) − γ (1 + B S ) (cid:21) . [97]Thus the two equations [96] and [97] are related according to | r S | d ˜ B/ d t = ( r S · )d B S / d t , and hence˜ B = [( r S · ) / | r S | ] B S . [98]The dynamics [96] conserves the energy E (cid:16) B S , dd t B S (cid:17) = 12 | r S | (cid:16) dd t B S (cid:17) + A S h β (1 + αB S ) − γ (1 + B S ) i [99]and we can use E ( B S , d B S / d t ) = E ( B S (0) , (d B S / d t )(0)) to obtaindd t B S = r(cid:16) dd t B S (0) (cid:17) + 2 A S | r S | h γ B S − β αB S − (cid:16) γ B S (0) − β αB S (0) (cid:17)i . [100]Let us assume that B S (0) = (d B S / d t )(0) = 0 then this simplifies todd t B S = r A S | r S | (cid:16) γ B S − β αB S − γ + β (cid:17) . [101]The argument of the square root above is non-negative if αβ/γ ≥ (1+ αB S ) / (1+ B S ) , [102]equivalently, if γ/ (1 + B S ) − β/ (1 + αB S ) − γ + β ≥
0. We note that for the B S = 0 and B S = ∞ this inequality reduces to αβ ≥ γ and β ≥ γ , respectively. The right hand side of [102] is monotonically increasing on the interval B S ∈ [ 0 , ∞ ) when α >
1, and monotonicallydecreasing if α <
1. Hence we need to satisfy β ≥ γ when α >
1, and αβ ≥ γ when α <
1. The RHS of [101] is a monotonic increasingfunction of B S when β/α > γ for α > , and αβ > γ for α < B S → ∞ in the right hand side of [101] gives usdd t B S = p A S | r S | ( β − γ ) [104]and hence B S ( t ) = p A S | r S | ( β − γ ) t + const . [105]If the above monotonicity condition [103] is satisfied, then B S ( t ) ≤ t/τ, [106]where τ is the time constant τ = 1 / p A S | r S | ( β − γ ) . [107]Furthermore, for α > B ∗ S = α ( β − γ ) + ( α − p αβγα ( αγ − β ) , [108]when β/α < γ ≤ β, [109] Alexander Mozeika et al.
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October 30, 2019 | vol. XXX | no. XX | R A F T B S ˙ B S At Figure 7: The (average) amount of antibodies, B S , rate ˙ B S and (normalised)antigen A , (top blue curve is the self-antigen and bottom red curve is the anti-gen) plotted as a function of time t for A S ( ) | r S | = 10, ↵ = 10, = 0 .
01 and = 0 . B S ! 1 in the RHS of (94) gives us˙ B S = q A S ( ) | r S | ( ) (98)and hence B S ( t ) = q A S ( ) | r S | ( ) t + const. . (99)If the condition in (97) is satisfied then B S ( t ) = c t, (100)where c = p A S ( ) | r S | ( ), is an upper bound for the solution of (94).Furthermore, for ↵ > B ⇤ S = ↵ ( ) + ( ↵ p ↵ ↵ ( ↵ ) , (101)when ↵ < , (102)which gives us the upper bound (100) with c = s A S ( ) | r S | ✓ (1 + B ⇤ S ) (1 + ↵B ⇤ S ) + ( ) ◆ . (103)We solve the equation (83) numerically in the regimes (97) and (102) for agiven , A S ( ) | r S | . The solutions of this equation are plotted in Figures 7–10.Also we compare the upper bound (100) with a solution of (83) in Figure 11.The upper bound allows us to summarise various regimes of the univalent Agbinding experiment in one ‘figure as follows.Let us now consider the normalised version A ( t t ) = 1 A ( b ( t )) ( t t ) Z t t A ( b (˜ t )) d˜ t, (104)26 Fig. 5.
The average Ab concentration, B S , and the rate ˙ B S = d B S / d t and (normalised) Ag A (top blue curve: self-Ag; bottom red curve: target Ag), shown as functions oftime t for A S | r S | = 10 , α = 10 , β = 0 . and γ = 0 . . B S ˙ B S At Figure 8: The (average) amount of antibodies, B S , rate ˙ B S and (normalised)antigen A , (top blue curve is the self-antigen and bottom red curve is the anti-gen) plotted as a function of time t for A S ( ) | r S | = 10, ↵ = 10, = 0 .
01 and = 0 . B S ˙ B S At Figure 9: The (average) amount of antibodies, B S , rate ˙ B S and (normalised)antigen A , (top blue curve is the self-antigen and bottom red curve is the anti-gen) plotted as a function of time t for A S ( ) | r S | = 10, ↵ = 1, = 0 .
01 and = 0 . A
1, of the damage (12) and the normalised version S ( t t ) = 1 A S ( b ( t )) ( t t ) Z t t A S ( b (˜ t )) d˜ t, (105)where 0 S
1, of (17). The latter gives us the (normalised) self-damage0 S
1. For the mean field system ( ?? ), on the time interval [0 , t ], abovegives us A ( t ) = 1 t Z t ↵B S (˜ t ) d˜ t (106)and S ( t ) = 1 t Z t B S (˜ t ) d˜ t. (107)We note that ↵B S ) is a monotonic decreasing function of B S and hence27 Fig. 6.
The average Ab concentration, B S , and the rate ˙ B S = d B S / d t and (normalised) Ag A (top blue curve: self-Ag; bottom red curve: target Ag), shown as functions oftime t for A S | r S | = 10 , α = 10 , β = 0 . and γ = 0 . . So here the time constant in [106] is different, and given by τ = 1 r A S | r S | (cid:16) γ B ∗ S − β αB ∗ S + β − γ (cid:17) . [110]We solve equation [96] numerically in the regimes [103] and [109], for a given values of β and A S | r S | . The solutions are plotted inFigures 5–8. Also we compare the upper bound [106] with a typical solution of [96] in Figure 9.Let us now consider the normalised damage per unit of time δ A ( t − t ) = 1 A ( b ( t )) ( t − t ) Z t t d t A ( b ( t )) , [111]where 0 ≤ δ A ≤
1, and a similar integral δ S ( t − t ) = 1 A S ( b ( t )) ( t − t ) Z t t d t A S ( b ( t )) , [112]where 0 ≤ δ S ≤
1, which defines the (normalised) self-damage per unit of time 1 − δ S , where 0 ≤ − δ S ≤
1. For the scenario describedby the equation [96], on the time interval [0 , t ], the above expressions give us δ A ( t ) = 1 t Z t d t
11 + αB S ( t ) , δ S ( t ) = 1 t Z t d t
11 + B S ( t ) . [113]Since 1 / (1 + αB ) decreases monotonically with B , from B S ( t ) < t/τ we obtain for the regime [103] the two lower bounds δ A ( t ) ≥ δ ∗ A ( t ) = ταt log (cid:16) α tτ (cid:17) δ S ( t ) ≥ δ ∗ S ( t ) = τt log (cid:16) tτ (cid:17) , [114]with the time constant τ − = p A S | r S | ( β − γ ) , [115]
11 + B S ( t ) . [113]Since 1 / (1 + αB ) decreases monotonically with B , from B S ( t ) < t/τ we obtain for the regime [103] the two lower bounds δ A ( t ) ≥ δ ∗ A ( t ) = ταt log (cid:16) α tτ (cid:17) δ S ( t ) ≥ δ ∗ S ( t ) = τt log (cid:16) tτ (cid:17) , [114]with the time constant τ − = p A S | r S | ( β − γ ) , [115] et al. R A F T B S ˙ B S At Figure 10: The (average) amount of antibodies, B S , rate ˙ B S and (normalised)antigen A , (top blue curve is the self-antigen and bottom red curve is the anti-gen) plotted as a function of time t for A S ( ) | r S | = 10, ↵ = 0 . = 0 .
01 and = 0 . B S A t
Figure 11: Left: The (average) amount of antibodies, B S , (blue line) and upperbound (black line) plotted as a function of time t for ↵ = 10 and = 0 . A , (the top ‘blue’ curves is self-antigen andthe bottom ‘red’ curves is antigen) and the lower bound plotted as a function oftime t for ↵ = 10 and = 0 . t for ↵ = 0 . = 0 . = 0 .
01 and A S ( ) | r S | =10. B S ( t ) = c t gives us the lower bounds A ( t ) ⇤ A ( t ) = 1 ↵c t log (1 + ↵c t ) (108)and S ( t ) ⇤ S ( t ) = 1 c t log (1 + c t ) , (109)with the ‘time’ constant c = q A S ( ) | r S | ( ) , (110)for the regime in (97).Let us consider the function ⇤ ( x ) = x log(1 + x ) in the domain x (0 , ).The derivative of this function ⇤0 ( x ) = x (1+ x ) log(1+ x )(1+ x ) x , by the inequality28 Fig. 7.
The average Ab concentrations , B S , and the rate ˙ B S and (normalised) Ag A (top blue curve: self-Ag; bottom red curve: target Ag), shown as functions of time t for A S | r S | = 10 , α = 1 , β = 0 . and γ = 0 . . B S ˙ B S At Figure 10: The (average) amount of antibodies, B S , rate ˙ B S and (normalised)antigen A , (top blue curve is the self-antigen and bottom red curve is the anti-gen) plotted as a function of time t for A S ( ) | r S | = 10, ↵ = 0 . = 0 .
01 and = 0 . B S A t
Figure 11: Left: The (average) amount of antibodies, B S , (blue line) and upperbound (black line) plotted as a function of time t for ↵ = 10 and = 0 . A , (the top ‘blue’ curves is self-antigen andthe bottom ‘red’ curves is antigen) and the lower bound plotted as a function oftime t for ↵ = 10 and = 0 . t for ↵ = 0 . = 0 . = 0 .
01 and A S ( ) | r S | =10. B S ( t ) = c t gives us the lower bounds A ( t ) ⇤ A ( t ) = 1 ↵c t log (1 + ↵c t ) (108)and S ( t ) ⇤ S ( t ) = 1 c t log (1 + c t ) , (109)with the ‘time’ constant c = q A S ( ) | r S | ( ) , (110)for the regime in (97).Let us consider the function ⇤ ( x ) = x log(1 + x ) in the domain x (0 , ).The derivative of this function ⇤0 ( x ) = x (1+ x ) log(1+ x )(1+ x ) x , by the inequality28 Fig. 8.
The average Ab concentrations, B S , and the rate ˙ B S and (normalised) Ag A (top blue curve: self-Ag; bottom red curve: target Ag), shown as functions of time t for A S | r S | = 10 , α = 0 . , β = 0 . and γ = 0 . . Let us consider the function δ ∗ ( x ) = x − log(1 + x ) for x ∈ (0 , ∞ ). Its derivative is δ ∗0 ( x ) = [ x − (1 + x ) log (1 + x )][(1 + x ) x ]. Dueto the inequality log (1 + x ) ≥ − (1 + x ) − , this derivative is negative for any finite x , so δ ∗ ( x ) is a monotonic decreasing function with δ ∗ ( x ) → x → δ ∗ ( x ) → x → ∞ . Since the image of δ ∗ ( x ) is the interval [0 ,
1] the function 1 − δ ∗ ( x ) is monotonic increasingon the same domain. It follows that δ ∗ A ( t ) → t →
0, implying that the (normalised) damage δ A ( t ) → δ ∗ A ( t ) → t → ∞ . Also 1 − δ ∗ S ( t ) → t →
0, implying that the self-damage 1 − δ S ( t ) → − δ ∗ S ( t ) → t → ∞ . For α = 1(where the strengths of antibody interaction with non-self and self are identical) we obtain δ ∗ A = δ ∗ S and the damage δ ∗ A (lower bound) islinearly related to the self-damage he self-damage 1 − δ ∗ S (upper bound) via δ ∗ A = 1 − (1 − δ ∗ S ). For α < δ ∗ A > − (1 − δ ∗ S ), so for a small reduction in the damage δ ∗ A we find a large increase in the damage to self 1 − δ ∗ S . For α > δ ∗ A < − (1 − δ ∗ S ), i.e. for a large reduction in δ ∗ A we have a small increase in 1 − δ ∗ S .We (re-)label the antibodies such that λ ≤ λ ≤ · · · ≤ λ M . We define the mean and the variance of the binding strengths toself-antigen, m ( r S ) = M − P Mµ =1 r Sµ and σ ( r S ) = M − P Mµ =1 ( r Sµ ) − ( M − P Mµ =1 r Sµ ) , and consider | r S | = M − P Mµ =1 λ − µ ( r Sµ ) .We note that for λ µ = λ : λ | r S | = σ ( r S ) + m ( r S ) , [116]Thus the time constant τ is given by 1 /τ ( λ ) = p A S ( ) λ − [ σ ( r S ) + m ( r S )] ( β − γ ) [117]Second, the weighted average M − P Mµ =1 λ − µ (cid:0) r Sµ (cid:1) , with λ − µ ≥ µ , is bounded from below by λ − M M − P Mµ =1 ( r Sµ ) and fromabove by λ − M − P Mµ =1 ( r Sµ ) . Hence the time constant in [115] is bounded according to τ ( λ ) ≤ τ ( λ ) ≤ τ ( λ M ) [118]This fact, in combination with the monotonicity of the x − log(1 + x ) as it appears in [114], gives us new lower bounds on the damage tonon-self and the damage on self: δ A ( t ) ≥ τ ( λ ) α t log (cid:16) α tτ ( λ ) (cid:17) δ S ( t ) ≥ τ ( λ ) t log (cid:16) tτ ( λ ) (cid:17) [119]We note that, since the time constant τ controls the speed of antigen removal, see equation [101], this speed is a monotonic increasing Alexander Mozeika et al.
PNAS |
October 30, 2019 | vol. XXX | no. XX | R A F T B S ˙ B S At Figure 10: The (average) amount of antibodies, B S , rate ˙ B S and (normalised)antigen A , (top blue curve is the self-antigen and bottom red curve is the anti-gen) plotted as a function of time t for A S ( ) | r S | = 10, ↵ = 0 . = 0 .
01 and = 0 . B S A t
Figure 11: Left: The (average) amount of antibodies, B S , (blue line) and upperbound (black line) plotted as a function of time t for ↵ = 10 and = 0 . A , (the top ‘blue’ curves is self-antigen andthe bottom ‘red’ curves is antigen) and the lower bound plotted as a function oftime t for ↵ = 10 and = 0 . t for ↵ = 0 . = 0 . = 0 .
01 and A S ( ) | r S | =10. B S ( t ) = c t gives us the lower bounds A ( t ) ⇤ A ( t ) = 1 ↵c t log (1 + ↵c t ) (108)and S ( t ) ⇤ S ( t ) = 1 c t log (1 + c t ) , (109)with the ‘time’ constant c = q A S ( ) | r S | ( ) , (110)for the regime in (97).Let us consider the function ⇤ ( x ) = x log(1 + x ) in the domain x (0 , ).The derivative of this function ⇤0 ( x ) = x (1+ x ) log(1+ x )(1+ x ) x , by the inequality28 Fig. 9.
Left: The average of Ab concentrations , B S , (blue line) and upper bound (black line) as a function of time t for α = 10 and γ = 0 . . Middle: The (normalised) Ag, A , (blue (cyan) curve is self-Ag and red (magenta) curve is target Ag) and the lower bound as a function of time t for α = 10 and γ = 0 . . Right: Ag and the lower bound(red (magenta) curve is target Ag and blue (cyan) curve is self-Ag) as a function of time t for α = 0 . and γ = 0 . . The β = 0 . and A S | r S | = 10 . function of the variance σ ( r S ) and the mean m ( r S ) of the vector of affinities r S , i.e. of the antibody repertoire. Thus, having a repertoirewith a higher variance facilitates a more rapid Ag removal.
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