Principia Biologica. Intelligently Deciphering Unintelligible Design. Bud Mishra. Professor of Computer Science, Mathematics and Cell Biology ¦ Courant Institute, NYU School of Medicine, Tata Institute of Fundamental Research, and Mt. Sinai School of Medicine. Robert Hooke.
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Professor of Computer Science, Mathematics and Cell Biology
¦
Courant Institute, NYU School of Medicine, Tata Institute of Fundamental Research, and Mt. Sinai School of Medicine
Hypotheses non fingo.I feign no hypotheses.Principia Mathematica.
first temporal derivative:
rate
second spatial derivative:
flux
¶a/¶t = Dar2a
a: concentration
Da: diffusion constant
reaction
diffusion

b
ReactionDiffusionTuring, A.M. (1952).“The chemical basis of morphogenesis.“ Phil. Trans. Roy. Soc. London B237: 37
d[A]/dt=F(1[A])
d[B]/dt=(F+k)[B]
reaction: d[A]/dt = d[B]/dt = [A][B]2
diffusion: d[A]/dt=DA2[A]; d[B]/dt=DB2[B]
¶ [A]/¶t = F(1[A]) – [A][B]2 + DA2[A]
¶ [B]/¶t = (F+k)[B] +[A][B]2 + DB2[B]
Reactiondiffusion: an exampleA+2B ! 3B
B ! P
B extracted
at rate F,
decay at rate k
A fed at rate F
Pearson, J. E.: Complex patterns in simple systems. Science261, 189192 (1993).
S = {A, T, C, G}
“The central dogma states that once `information' has passed into protein it cannot get out again. The transfer of information from nucleic acid to nucleic acid, or from nucleic acid to protein, may be possible, but transfer from protein to protein, or from protein to nucleic acid is impossible. Information means here the precise determination of sequence, either of bases in the nucleic acid or of amino acid residues in the protein.”
Transcription
Translation
DNA
RNA
Protein
Terminator
1035bp
Transcriptional
Initiation
Transcriptional
Termination
Gene
RNA, Genes and Promoters“Work on the mathematics of growth as opposed to the statistical description and comparison of growth, seems to me to have developed along two equally unprofitable lines… It is futile to conjure up in the imagination a system of differential equations for the purpose of accounting for facts which are not only very complex, but largely unknown,…What we require at the present time is more measurement and less theory.”
en
en
wg
WG
WG
ptc
PTC
PTC
+
PH
PH
cid
CID
CN
hh
HH
HH
Celltocell
interface
a neighbor
a cell
positive
interacions
negative
interacions
mRNA
proteins
The Network of InteractionThe SYSTEMS BIOLOGY
Systems BiologyCombining the mathematical rigor of numerology with the predictive power of astrology.
Cyberia
Numerlogy
Astrology
Numeristan
HOTzone
Astrostan
Infostan
Interpretive Biology
Computational Biology
Integrative Biology
Bioinformatics
BioSpice
We claim that, by drawing upon mathematical approaches developed in the context of dynamical systems, kinetic analysis, computational theory and logic, it is possible to create powerful simulation, analysis and reasoning tools for working biologists to be used in deciphering existing data, devising new experiments and ultimately, understanding functional properties of genomes, proteomes, cells, organs and organisms.
Simulate Biologists! Not Biology!!
Biology of the future should only involve a biologist and his dog: the biologist to watch the biological experiments and understand the hypotheses that the dataanalysis algorithms produce and the dog to bite him if he ever touches the experiments or the computers.
Canonical Form:
Glycogen
P_i
Glucose
Glucose1P
Phosphorylase a
Phosphoglucomutase
Glucokinase
Glucose6P
Phosphoglucose isomerase
Fructose6P
Phosphofructokinase
LacI, tetR & l cI
Arranged in a cyclic manner (logically, not necessarily physically) so that the protein product of one gene is rpressor for the next gene.
LacI!:tetR; tetR! TetR
TetR!:l cI; l cI !l cI
l cI!:lacI; lacI! LacI
An Artificial ClockLeibler et al., Guet et al., Antoniotti et al., Wigler & Mishra
x1
x2

x3
x4

x5
x6
Cascade Model: Repressilator?dx2/dt = a2 X6g26X1g21  b2 X2h22
dx4/dt = a4 X2g42X3g43  b4 X4h44
dx6/dt = a6 X4g64X5g65  b6 X6h66
X1, X3, X5 = const
G0
G0
G0
G0
G0
G0
G0
[b]
G1
G1
G1
G1
G1
G1
M
M
S
S
S
S
S
G2
G2
G2
[a]
M
M
M
M
M
Computation Tree¼ {X’  X’ = X + F(X,0) h + d, d < e},
for a suitably chosen
e´ F’(X,0) h2/2! + F’(X,0) h3/3! + L 
Physically adjacent cells laterally inhibit each other’s ciliation (Delta production)
Rewriting…
Where:
Collier et al.
Q1 mathematical model of DeltaNotch intercellular signalling
Both OFF
Q2
Delta ON
Q3
Notch ON
Q4
Both ON
OneCell DeltaNotch Hybrid AutomatonGhosh et al.
Q1 mathematical model of DeltaNotch intercellular signalling
Neither
Q2
Delta
Q2
Delta
Q1
Neither
X
Q3
Notch
Q4
Both
Q3
Notch
Q4
Both
TwoCell DeltaNotch SystemCell 1
16 Discrete States
Cell 2
Q3 mathematical model of DeltaNotch intercellular signalling
Notch
Q2
Delta
X
Q2
Delta
Q3
Notch
X
System Properties:True & ApproximateQ2 Delta mathematical model of DeltaNotch intercellular signalling
Q3 Notch
X
Impossibility Of Reaching Wrong Equilibrium:Step 1. Formally encode the behavior of the system as a hybrid automaton
Step 2. Formally encode the properties of interest in a powerful logic
Step 3. Automate the process of checking if the formal model of the system satisfies the formally encoded properties using Model Checking
Time is Linear
Time is Branching
For p mathematical model of DeltaNotch intercellular signallingAP:
s ² p p L(s)s ²p p L(s)
s ² f Æ gs ² fand s ² g
s ² f Ç g s ² fors ² g
s ² EX f =hs0s1... i from ss1² f
s ² E(f U g) = hs0s1... i from s
j0 [sj² gand i : 0 i j [si² f] ]
s ² EG f = hs0s1... i from s i 0: si² f
Semantics for CTLVariation of the initial concentration of PRPP does not change the steady state.(PRPP = 10 * PRPP1) implies steady_state()
This query will be true when evaluated against the modified simulation run (i.e. the one where the initial concentration of PRPP is 10 times the initial concentration in the first run – PRPP1).
Persistent increase in the initial concentration of PRPP does cause unwanted changes in the steady state values of some metabolites.
If the increase in the level of PRPP is in the order of 70% then the system does reach a steady state, and we expect to see increases in the levels of IMP and of the hypoxanthine pool in a “comparable” order of magnitude. Always (PRPP = 1.7*PRPP1) implies steady_state()
QueriesTRUE
TRUE
Consider the following statement: change the steady state.
Eventually
(Always (PRPP = 1.7 * PRPP1)impliessteady_state()and Eventually
Always(IMP < 2* IMP1))and Eventually (Always (hx_pool < 10*hx_pool1)))
where IMP1 and hx_pool1 are the values observed in the unmodified trace. The above statement turns out to be false over the modified experiment trace..
In fact, the increase in IMP is about 6.5 fold while the hypoxanthine pool increase is about 60 fold.
Since the above queries turn out to be false over the modified trace, we conclude that the model “overpredicts” the increases in some of its products and that it should therefore be amended
QueriesFalse
Alur et al,
Invariant: False change the steady state.
Flows: Null
Mandelbrot Hybrid AutomatonLet:
Then:
Reachability Query:
Damned Doggs.
Vindica me deus.
Sir Nicholas Gimcrack character in
The Virtuoso, a play by Thomas Shadwell.
The end… change the steady state.