the immune system from a statistical mechanics...
TRANSCRIPT
Performance and breakdown of the immune system from a
statistical mechanics perspective
Elena Agliari Università di Parma
INFN - Gruppo Collegato di Parma
Bari, SM&FT, 21-23 September 2011venerdì 23 settembre 2011
Statistical Mechanics and Theoretical Immunology
Immune system devoted to protect host body against damaging substances
Made up of a huge number of different kinds of cells and messengers, which must be properly
orchestrated to ensure safe performance
Pattern recognition, memory storage, self/non-self discrimination are cooperative features
Statistical mechanics Systemic viewpoint to evidence emergent
properties and the key mechanisms underlying (mis)functioning of the system
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B cells make up a “ferromagnetic network”Experimental facts
B lymphocytes are dichotomic →σ = ± 1Each lymph. carries a specificity →ξ={1,0,...,1}
Clone (ξi) ↔ Specific antibody (ξi)Several (N) different clones → {ξ1, ξ2,..., ξN}
Model 0System of NxM spins
N order parameters: mi(t) = Σa σia(t)/M → ci(t) ∼ exp[mi(t)/τ]
Antigenic stimulation ↔ External local field hi
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Burnet’s (Nobel Prize 1960) clonal expansion theory
H0 = - cAg Σi hi mi
cAg is antigenic concentration
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Model 1System of NxM spins
N order parameters: mi(t) = Σa σia(t)/M → ci(t) ∼ exp[mi(t)/τ]Exp. findings & “Small” repertoire → Functional network
Model 0System of NxM spins
N order parameters: mi(t) = Σa σia(t)/M → ci(t) ∼ exp[mi(t)/τ]Antigenic stimulation ↔ External local field hi
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H0 = - cAg Σi hi mi
cAg is antigenic concentration
H1 = - 1/N Σi,j Jij mi mj
Jij measures affinity between clones i and j
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Jerne’s (Nobel Prize 1984) idiotypic network theory
[Abs also function as internal images of Ags]
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The two approaches are synergic
H = H0 + H1 = - cAg Σi hi mi - 1/N Σi,j Jij mi mjArbitrary couple of nodes, i, j, the µ-th entries of corresponding stringsare complementary iff !µ
i != !µj " # complementary entries c # [0, L]
cij =!L
µ=1[!µi (1$ !µ
j ) + !µj (1$ !µ
i )] =!L
µ=1[!µi + !µ
j $ 2!µi !µ
j ]
Attractive vs Repulsive interaction! weight ! " 0f!,L("i, "j) # !cij $ (L$ cij) % [$L, !L]Nodes can interact pairwise according to a couplingJij ! !(f!,L(!i, !j)) exp[f!,L(!i, !j)]/J" Jij = exp[cij(" + 1)# L], whenever cij > L/(" + 1), otherwise Jij = 01/J normalizing pre-factor ensuring Jij has finite, unitary average $", L
Attractive vs Repulsive interaction → weight α ≥ 0
# Complementary matches
Minimal assumption & Uncorrelation among nodes! uniform distribution for !µ
i ’s extraction P (c) =!
12
"L !Lc
"
Probability that Jij > 0, i.e. i " j, is p!,L =#L
c=!L/(!+1)"+1 P (c)
Coupling distribution P (Jij) =$
(1/2)L! L(Jij+L)/(1+!)
"if J > 0
1# p if Jij = 0
Coordination number: zi =!N
j=1 Aij =!N
j=1!Jij"
Weighted coordination number: wi =!N
j=1 Jij
Coordination number: zi =!N
j=1 Aij =!N
j=1!Jij"
Weighted coordination number: wi =!N
j=1 Jij
Coupling distribution
Coordination number
Weighted Coordination number
➪ Ferromagnetic Correlated-Random-Bond Net
Idiotypic network [Varela et al.]
Cumulative frequency distributions of significant interactions (normalized optical densities) measured between a collection of 435 Abs [Carneiro et al. (1996)]
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Cooperative features of the idiotypic network
- Low-Dose Tolerance- Bell-shape response- Memory Effects
Surviving self-reacting clones are quiescent due to the interaction with the network (over which information is spread)
Negative selection in bone marrow: most self-reacting clones are deleted
High weighted connectivity (low reactivity) group ➙ self addressed
Low weighted connectivity (high reactivity) group ➙ non-self addressed
Self-NonSelf Discrimination
Interaction matrix [Kearney et al.]venerdì 23 settembre 2011
Cooperative features of the idiotypic network
- Low-Dose Tolerance- Bell-shape response- Memory Effects
Surviving self-reacting clones are quiescent due to the interaction with the network (over which information is spread)
Negative selection in bone marrow: most self-reacting clones are deletedSelf-NonSelf Discrimination
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
a
corre
latio
n co
effic
ient
Self=50Self=100Self=500Self=1000Self=5000
Simulation of negative selection yields to correlation between w and self-affinity
Negative selection is automatically accounted in our networkvenerdì 23 settembre 2011
Helper cells make the system complex
B lymphocytes. Play as APC
. Make specific Abs. Need signal of chemical
messengers from Th
Tk lymphocytes. Induce death of dangerous cells. Need signal of chemical messengers from Th
Th lymphocytes. No effector functions. Receive first signal from APC. Release/Absorb regulatory agents (cytokines)
Cytokines. Auto-, Para-, Endo-
crine effects. Induce differentiation/
growth/death. Balance is crucial!
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H
P (ki)
! !
P (bi)
bi
k0
hi
BK
ki
b0
Helper cells make the system complex
B lymphocytes. Play as APC
. Make specific Abs. Need signal of chemical
messengers from Th
Tk lymphocytes. Induce death of dangerous cells. Need signal of chemical messengers from Th
Th lymphocytes. No effector functions. Receive first signal from APC. Release/Absorb regulatory agents (cytokines)
Cytokines. Auto-, Para-, Endo-
crine effects. Induce differentiation/
growth/death. Balance is crucial!
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H
P (ki)
! !
P (bi)
bi
k0
hi
BK
ki
b0
Th lymphocytesEach clone hi = ±1, i = 1,...,HActual concentration ∼ Σi exp(hi)
B lymphocytesSelf-regulatory effectsEach clone bi =N(0,1), i = 1,...,B
Actual concentration ∼ Σi exp(bi)
Tk lymphocytesSelf-regulatory effectsEach clone ki =N(0,1), i = 1,...,K
Actual concentration ∼ Σi exp(ki)Cytokinesξiμ cytokine acting between hi and bμ (kμ), assumed symmetric
ξiμ = ± 1 for excitatory/inhibitory stimulation, P(ξiμ = +1) = P(ξiμ = -1) = 1/2{ξ} quenched and encodes proper “strategies”
3-partiteSG
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Partition function:
Gauss-integrate
n-partite spin-glass Sum of n-1 independent neural networks
Helpers promote/suppress, via cytokines, the effectors branches
Cytokines directly connect helpers via Hebbian interaction, making them able to learn, store, retrieve patterns
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1
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1 2
3 4
-1
-1-1
+1
+1
+3
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1 2
3 4
-1
-1-1
+1
+1
+3
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Retrieval pattern ξ1 ⇔ Stimulation b1
1
2
3As a result of a learning process, cytokines
ensures the dynamic stability of certain helper configuration, corresponding to
stimulation of certain bi
Mattis magnetization for the μ-th state
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Scaling for subset size (high load)α, γ standard hematological markers
As the amount of patterns increases, the network fails retrieval
CD4/CD8 <1 → HIV progression in AIDS “Inverted ratio” reduces lymphocyte activity and responsiveness to foreign antigens
System actually always subjected to external stimuli (viruses, bacteria, tumor cells) on effector branches → positive mean activity k0 [polyclonal expansion]
Random field → N(0,1), TDL
consistent with low-titer self-Abs
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β noise levelΦ rescaled measure of mean activityα/γ lymphocyte ratios
Ordered work (organized effector activation) [∼ t]→ introduction of disorder within the system [∼ √t]
0
0.2
0.4
0.6
0.8
1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
1/!
"
0
0.2
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1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
1/!
"
0
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1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
1/!
"
minimize RS free-energy w.r.t. order parameters
self-consistent equations for
Φ=0 Φ=0.5 Φ=1
SG
RetrievalPureState
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Emergence of pathologies as retrieval failure
Abnormal lymphocyte accumulation (e.g. ALPS, leukemia)
→ severe autoimmune phenomena
Φ>>1 RF phase prevails against retrieval phase
HIV to overt AIDS[HIV infect and kill h]EBV [immortalize B cells]
→ populations imbalance yields reduced immune functions and responsiveness
SG emergence and retrieval failureSlow noise prevails
Free-radicals, cholesterol, damage accumulation, debris from lysis→ slow down in recognition processes and speed up other pathologies
PM emergence and retrieval failureFast noise prevails
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Recovering the system as a dynamical process
Agreement between idiotypic and tripartite
approaches
Agreement between systemic and
dynamical approaches
Ornstein-Uhlenbeck process
Mean-Field approximationsassuming 〈J〉 exists strictly positive
Gaussian effectors activity with variance given by weighted connectivity
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Getting close to biology
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Cytokines{ξ} quenchedExcitatory/inhibitory/absent stimulation
P(ξiμ = +1) = P(ξiμ = -1) = (1-d)/2P(ξiμ = 0) = d
Finite degree of dilution h-h still FC Network more effectiveParallel retrieval occurs Critical parameters are correctedIntermediate dilutions SG phase emergesExtremely high dilution h-h network diluted
0
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1
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1.4
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1.8
-2 -1 0 1 2
Campi alla Barra T=0 Alfa=0.05
dil_0.0dil_0.1dil_0.2
dil_0.25dil_0.3
dil_0.35dil_0.4
dil_0.45dil_0.5dil_0.6dil_0.7dil_0.8
Field distributions
Getting close to biology
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- Unifying picture including Burnet and Jerne theories
- Idiotypic network based on eliciting, complementarity-based couplings recovers basic facts in immunology
- Self/Non-self discrimination merges experimental results and ontogenesis features
- Learning and Memory needs complex interactions yielded by a “central control” provided by Th
- Errors ≠ Spurious states → Realize overlap of several strategies hence response hires several correlated lymphocytes
- Auto-immune diseases related to a few main sources of “noise”
- The tripartite system is consistent with idiotypic-network as well as dynamical approaches
Conclusions
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Work in collaboration with:Adriano Barra, Francesco Guerra, Francesco Moauro
(Sapienza, Università di Roma)Raffaella Burioni, Aldo Di Biasio (Università di Parma)
Silvio Franz, Thiago Sabetta (Université Paris-Sud, Orsay)
List of related publications:
EA, A. Barra, A Hebbian approach to complex network generation, EPL (2011)
EA, A. Barra, F. Guerra, F. Moauro, A statistical mechanics on ALPS, J. Theor. Biol. (2011)
A. Barra, EA, Solving statistical mechanics on complex topologies trough neural network
techniques, JStat (2011)
EA, A. Barra, F. Guerra, K. Gervasi-Vidal, Associative Learning as origin of CFS, submitted (2011)
EA, A. Barra, Statistical mechanics of idiotypic immune networks, submitted (2011)
A. Di Biasio, EA, A. Barra, R. Burioni, Mean-field cooperativity in chemical kinetics, Theor.
Chem. Acc. (2011)
A. Barra, EA, A. Galluzzi, F. Guerra, F. Moauro, Parallel Processing Immune Networks, (2011)A. Barra, S. Franz, T. Sabetta, EA, Ontogenesis in B-cell and self-discrimination, (2011)
EA, C. Cioli, E. Guadagnini, Percolation on correlated random networks, PRE (2011)
A. Barra, EA, A Statistical Mechanics approach to autopoietic immune networks, JStat (2010)venerdì 23 settembre 2011
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Cooperative features of the idiotypic network
Low-Dose Tolerance c hik > wk
N
In “healthy state”, antigenic concentration to elicit response is
lower-bounded
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Cooperative features of the idiotypic network
Bell-shape response
0 500 1000 1500 2000−1
−0.5
0
0.5
1
1.5
2
t
mi,
mi,
x
Ab1
AAb1
Antigen
0 2000 4000 6000 8000
−0.5
0
0.5
1
tm
i,m
i,m
j,m
j,c
x
Ab1
AAb1
Ab2
AAb2
Antigen
First and second best matching antibody reacting with the antigen
Pathological example of a chronic response
Low-Dose Tolerance
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Cooperative features of the idiotypic network
Memory Effects
Antigen ! !1(Ab) ! !2(Ab) ! !3( ¯Ab)...Circuit of length l: " = {i1, i2, ..., il}Overall strength J! = l!1
!lk=1 Jik,ik+1 ! intrinsic robustness of "
Overall weight w! = l!1!l
k=1 wik ! influence of ext environment on "
Excited self-reinforced loops
Antigen ! !1(Ab) ! !2(Ab) ! !3( ¯Ab)...Circuit of length l: " = {i1, i2, ..., il}Overall strength J! = l!1
!lk=1 Jik,ik+1 ! intrinsic robustness of "
Overall weight w! = l!1!l
k=1 wik ! influence of ext environment on "
Antigen ! !1(Ab) ! !2(Ab) ! !3( ¯Ab)...Circuit of length l: " = {i1, i2, ..., il}Overall strength J! = l!1
!lk=1 Jik,ik+1 ! intrinsic robustness of "
Overall weight w! = l!1!l
k=1 wik ! influence of ext environment on "
Antigen ! !1(Ab) ! !2(Ab) ! !3( ¯Ab)...Circuit of length l: " = {i1, i2, ..., il}Overall strength J! = l!1
!lk=1 Jik,ik+1 ! intrinsic robustness of "
Overall weight w! = l!1!l
k=1 wik ! influence of ext environment on "
100 102 1040
1000
2000
w!
10−6 100 1050
200
400
600
800
J!
l = 4, Self
l = 4, Non-Self
l = 3, Self
l = 3, Non-Self
The less diluted the network the smaller the region where memory effects may emerge
Region centered on <k> = O(1012) → L = O(102) consistent with experiments
Bell-shape responseLow-Dose Tolerance
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Cooperative features of the idiotypic network
Robustness under Ab correlation
Hypersomatic mutations in germinal centersEvolution not completely random Selective specification
➪ P(ξ=±1)= (1±a)/2a ∈ [-1,+1]
(ageing)
Correlation first induces fringes and then breaks down connectivity
- Low-Dose Tolerance- Bell-shape response- Memory Effects- Self-NonSelf Discrimination
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N!10000, L!64, a!0.570000, "!0.700000.9098 nodi isolati.
N!10000, L!64, a!0.590000, "!0.700000.9472 nodi isolati.
N!10000, L!64, a!0.610000, "!0.700000.9787 nodi isolati.
N!10000, L!64, a!0.630000, "!0.700000.9896 nodi isolati.
Small (w.r.t. ER) frequency of short
loops in underpercolated regime
0.35 0.4 0.45 0.5 0.55 0.6 0.650
2000
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10000
a
#
0.35 0.4 0.45 0.5 0.55 0.6 0.650
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a
#
0.35 0.4 0.45 0.5 0.55 0.6 0.6510−5
100
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1010
a
#
Scan di a. N=10000, L=64; !=0.7; 10 prove.
isolated nodes dimerstrimers3stars
triangles quadrangles trianglesquadrangles
0.35 0.4 0.45 0.5 0.55 0.6 0.650
2000
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6000
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10000
a
#
0.35 0.4 0.45 0.5 0.55 0.6 0.650
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#
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0
0.5
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0.35 0.4 0.45 0.5 0.55 0.6 0.650
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a
#
0.35 0.4 0.45 0.5 0.55 0.6 0.650
1
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5x 108
a
#
0.35 0.4 0.45 0.5 0.55 0.6 0.6510−5
100
105
1010
a
#
Scan di a. N=10000, L=64; !=0.7; 10 prove.
isolated nodes dimerstrimers3stars
triangles quadrangles trianglesquadrangles
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H
P (ki)
! !
P (bi)
bi
k0
hi
BK
ki
b0
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Standard (Φ=0) phase diagramfor Hopfield model
P: paramagnetic phase, m=q=0SG: spin-glass phase, m=0, q≠0
F: pattern recall phase (pure states minimize f), m≠0, q≠0M: mixed phase (pure states are local but not global minima)
Tg: second order, TM and Tc: first order
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0.2 0.4 0.6 0.8 1.0m
0.2
0.4
0.6
0.8
1.0
q
0.2 0.4 0.6 0.8 1.0m
0.2
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0.6
0.8
1.0
q
0.2 0.4 0.6 0.8 1.0m
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1.0
q
0.2 0.4 0.6 0.8 1.0m
0.2
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0.8
1.0
q
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