CDAC 2018 Mishra immune system part a

Game Theoretic Models Biological Deception Multi-Cellularity Immune System
Immune Systems
Part A: Game Theory
Bud Mishra
Courant, NYU
Como, May 2018

Outline
1 Game Theoretic Models
Signaling Games
Nash Equilibria
Deception
2 Biological Deception
Examples of Deception in Biology
An Exhaustive Classiﬁcation of Deception in Biology
3 Multi-Cellularity
Geoﬀry-Cuvier Debate
Cellularization: Aligning Utility
Cancer & Cellular Deception
4 Immune System
Overall Picture
Innate System
Adaptive System

Self & Self-Reference
What contains
what?
Self, Encoding of
Self and
Recognition of Self
What cells know
that they belong to
a multi-cellular
organism?

Signaling Games
Two player games with incomplete information...
One player is informed ... the other player is not...
1 The informed player’s strategy set consists of signals
contingent on information
2 Uninformed player’s strategy set consists of actions contingent
on signals
Spence 1973, Zahari 1977, Lewis 2002, Sobel 2009

The Model of Signaling Games
Two players:
S − Sender (Informed)
R − Receiver (Uninformed)
Roles can be shared - partial information, distributed actions
TYPE: Random variable t whose support is given by T
(known to Sender S). π(·) = Probability distribution over T
is a prior belief of R that the sender’s type is t.

Game
Game
1 Player S learns t ∈ T
2 S send to R a signal s ∈ M.
3 R takes an action a ∈ A.
Payoﬀ function
ui∈{S,R}
: T × M × A → R.

Equilibrium
Behavior Strategies:
1 For S a function µ : T × M → [0, 1] such that
s∈M
µ(t, s) = 1, for all t.
µ(t, s) = Probability that S with type t sends signal s.
2 For R a function α : M × A → [0, 1] such that
a∈A
α(s, a) = 1, for all s.
α(s, a) = Probability that R takes action a following signal s.
Subjective probability.
β(t, s) =
µ(t, s)π(t)
t ∈T µ(t , s)π(t )
.

Proposition
Behavior strategies (α∗, µ∗) form a Nash equilibrium iﬀ for all t ∈ T
µ(t, s) > 0 implies
a∈A
US
(t, s, a)α(s, a)
= max
s ∈S
a∈A
US
(t, s , a)α(s , a);
& for all s ∈ S (s.t. t∈T µ(t, s)π(t) > 0)
α(s, a) > 0 implies
t∈T
UR
(t, s, a)β(t, a)
= max
a ∈A
t∈T
UR
(t, s, a )β(t, a ).

Signaling Games in Nature
Mapping Types and Actions into Signals:
f S
: T → A; f R
: A → T.
Sender
US
= I(T, M) + λS dS
(f S
(t), a).
Receiver
UR
= I(A, M) + λRdR
(t, f R
(a)).

Subjective Reality
Note that the distribution of signals received by R is given by
the probability distribution πM, where
πM(s) =
t∈T
πT (t)µ(s|t),
And the distribution of actions produced by R is given by the
probability distribution πA, where
πA(a) =
s∈M
πM(s)α(a|s).
Clearly πT and πA are probability distributions on T and A
respectively. If ˆπT is the probability distribution on T induced
by πA under the function fR, then
ˆπT (·) := πA(f −1
R (·)). (1)

Deception
A natural choice of measure for deception is given by the
relative entropy between the probability distributions πT and
ˆπT :
Deception := Rel. Entropy(ˆπT |πT )
=
t∈T
ˆπT (t) log2
ˆπT (t)
πT (t)
. (2)
This deﬁnition describes deception from the point of view of
the receiver. To get the notion of deception from the point of
view of the sender, one needs to play the game several rounds.

Signaling Games
Separating Equilibrium: Each type t sends a diﬀerent signal
Mt. f S : t → a[Mt]...
Pooling Equilibrium: All types t send a single signal s∗ with
probability 1.
Convention & Deception: The divergence between the
objective probabilities and the subjective probabilities induced
by conventional equilibria.
Solution: Costly Signaling; Credible and Non-credible threat;
Aligned Utilities; 2 + m + n players – m Recommenders + n
Veriﬁers

Recommenders and Veriﬁers
Recommenders aim
for Liveness
Veriﬁers aim for
Safety

Recommenders and Veriﬁers
Liveness
∀Sender,S (Type t ∈ T & Signal s ∈ M) ∃Receiver,R(Action a ∈ A)
AG[s → Fa US
(t, s, a) > θ∗
S ]
For every Uber driver with 4 or more stars, there is a passenger
with 4 or more stars waiting within 4 minutes for a ride.
Safety
∀Receiver,R(action a ∈ A) ∀Sender,S (Type t ∈ T & Signal s ∈ M)
AG[¬a U s UR
(t, s, a) > θ∗
R]
For every passenger with 4 or more stars, there is no need to
wait for more than 4 minutes to get a Uber driver with 4 or
more stars.

Thermo-Tolerance Game
An example of a biochemical Lewis signaling game is provided
by thermotolerance via RheA-HSP18 system:
RheA is a thermosensor, thus an informed agent; but HSP18
is needed to modulate thermotolerance, thus an uninformed
agent, which nonetheless acts in response to the signal
consisting of RheA’s conversion to a nonDNA-binding form.
They enable the cell to survive spikes in temperature in the
environment, thus improving the Shapley value of all the
macromolecules contained in the cells – including RheA and
HSP18.

Evolutionary transi-
tion
Type of Cooperation Signaling convention Type of subversion
Replicators Between monomer
subunits, forming
polymers that pro-
mote common inter-
est of the monomer
constituents
The first replicators
are unknown, but
would have utilized
molecular specificity
Parasitism of replicator func-
tion may have occurred by other
polymers
Protein translation Between mRNAs and
the ribosome
Genetic code ‘Deceiver’ tRNAs (tRNAs that
alter the genetic code e.g. sup-
pressor tRNAs) ‘Deceiver’ mR-
NAs (mRNAs that benefit the
sender gene, but not the host
genome)
Eukaryogenesis Between nucleus and
mitochondrion
Nuclear targeting sig-
nals, mitochondrial
targeting signals
Bacteria can use nuclear lo-
calization signals to gain entry
to the nucleus (a Trojan horse
strategy)
Between mRNA and
the spliceosome
Intron splice sites Selfish elements can ‘hide’ in
or mimic introns (see Table 1),
viruses sequester the splicing
machinery to regulate their gene
expression
Between DNA and hi-
stones
Histone code Some bacteria appear to modify
the histone code

Sexual reproduction Between two genders Species specific
chemical, visual and
auditory signaling
Many examples of deception due
to sexual conflict
Between two gametes Gamete fusion in-
volves the HAP2
protein. Sperm-egg
recognition is species
specific
Undescribed
Between two homolo-
gous chromosomes
Homologous recom-
bination is initiated
by Spo11
B chromosomes mimic sex chro-
mosomes leading to chromoso-
mal drive. Some examples of
meiotic drive utilize molecular
deception
Multicellularity Between cells Has arisen a num-
ber of times indepen-
dently – facilitated by
the evolution of cell
adhesion and signal-
ing, and immune sys-
tems
Cancers utilize a variety of
molecular mechanisms that dis-
rupt normal cell-cell recognition,
and evade the immune system
Eusociality Between related indi-
viduals
Has arisen a num-
ber of times indepen-
dently. Each has
established different
signaling conventions
based on acoustic, vi-
sual and chemical sig-
nals
Mimicry of acoustic signals and
pheromones
Humanity Between unrelated in-
dividuals
Spoken language Lying

An Old Debate – Goethe
Protagonists: Biologist Étienne Geoffroy St Hilaire (1772 – 1844)
and Anatomist Georges Cuvier (1769–1832)
Time & Place: French Academy – 1830.
Debate between Philosophical Anatomy (Geoffroy) vs. Empirical
Anatomy (Cuvier)
Geoffroy argued for the unity of animal kingdom.
Cuvier argued for the existence of four ‘embranchments’
vertebrates, arthropods, molluscs and echinoderms.
Who was right?

Consider the Lobster
Geoffroy’s Conjecture: The vertebrates were arthropods
upside-down; flip the dorsal-ventral morphogenesis.
An Anatomical Hurdle: It seems impossible to map
arthropod’s ventral nerve cord to vertebrates’ dorsal system.
It will be few centuries before we’d understand the role of
genes in the development via control of morphogen gradient.
Two genes sog and dpp (in arthropods) are flipped to the
homologous pairs chordin and bmp (in vertebrates).
sog in the fly, Drosophila, determines ventral development:
chordin in the toad, Xenopus , determines dorsal development.

Evo-Devo: Cell-level
We work with an abstraction of a cell that:
can (noisily) sense some kind of local information ι ∈ I,
depending on the application
can produce signals from a set S
can sense its environment e ∈ E = NS, i.e., the multiset of
signals produced by its neighbors
has a state σ ∈ Σ, some form of (bounded) memory
can perform actions a ∈ A depending on the application
A strategy is a mapping s : Σ × I × E → Σ × 2S × A.
We denote the individual components as sΣ, sS and sA.

Segmentation: Some screenshots of an organism
A one-dimensional organism after 0 & 100 steps. I is normalized to [0, 1]. The green
line shows the morphogen gradient as currently sensed by the cells (i.e., noisy). The
remaining lines depict the currently produced signals. Shaded regions depict sS after
bias has been applied.

Segmentation: More screenshots of an organism
... after 10000 & 15000 steps.

Equilibria
Separating Equilibrium: Each type t sends a different signal
Mt. f S : t → a[Mt]...
With two signals {BMP, Anti-BMP}, Arthropods
(Protostomes) and Vertebretes (Deuterostomes) represent two
different separating equilibria. Just as Geoffroy thought!
Pooling Equilibrium: All types t send a single signal s∗ with
probability 1
Are there examples of pooling equilibria in nature (on earth or
some other exoplanet)?

Hemichordates
Saccoglossus kowalevskii
(with a diﬀused CNS;
nerve-nets) - Considered a
Deuterstome

Signaling Equilibria
Phylogeny?

Shifting Equilibria
All the
Renegade Cells
Shifting
Balance:
Growth,
Proliferation,
Anoxia, Immune
Resistance,
Immortality,
Metastasis...

Sender and Receivers
Sender =
Dendritic Cells
Receiver =
Macrophage
Recommenders
= B Cells
Veriﬁers = T
Cells

Senders (DC) and Receivers (Macrophage)
Signaling with Cytokines; Acting by phagocytosis
Diﬀerent cytokines are produced depending on the type of
dendritic cell (DC) involved.
For example, the plasmocytoid dendritic cells can produce
high levels of type 1 interferons (signal), which leads to the
recruitment of another type of APC called a macrophage that
can engulf and destroy (action) pathogens.
DC’s also release cytokines that promote inﬂammation such
as IL-12 and TNF-α.

Safety: Educating the Verifiers (T Cells)
The dendritic cells (DC) capture antigens from invading
bodies, which they process and then present on their cell
surface, along with the necessary accessory or co-stimulation
molecules.
Due to this ability to process and present antigens, dendritic
cells are referred to as antigen presenting cells (APCs).
Although all dendritic cells can perform antigen presentation
to stimulate na¨ıve T cells, the different types of dendritic cells
have distinct markers and different locations.
In addition to their ability to activate na¨ıve T cells, dendritic
cells play a role in aiding regulatory T cell differentiation and
the development of T cell tolerance.

Liveness: Learning by Recommenders (B Cells)
Variations: B cells explore recommendations (obliviously) by
undergoing V(D)J recombination as they develop.
Positive selection occurs through antigen-independent
signaling involving both the pre-BCR and the BCR. If these
receptors do not bind to their ligand, B cells do not receive
the proper signals and cease to develop.
Negative selection occurs through the binding of self-antigen
with the BCR; If the BCR can bind strongly to self-antigen,
then the B cell undergoes one of four fates: clonal deletion,
receptor editing, anergy, or ignorance (B cell ignores signal
and continues development).
Memory: Upon antigen binding, the memory B cell takes up
the antigen through receptor-mediated endocytosis, degrades
it, and presents it to T cells as peptide pieces in complex with
MHC-II molecules on the cell membrane.
Interaction between Macrophage and B Cells (?)

End
La ﬁn
Die Ende
Shuryou
Slutten
Wakas
Sfarsit
Samapta
El ﬁn
Son
Ukuphela

CDAC 2018 Mishra immune system part a

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