Biochemical Programs and Mixed Analog-Digital Algorithms in the Cell

Biochemical Programs and Mixed Analog-Digital Algorithms in the Cell

François Fages

Inria Saclay – Ile de France EPI Lifeware

France

http://lifeware.inria.fr/

Outline

1. Formal biochemical reaction systems as a programming language

2. Functional specifications: example of the MAPK signaling network module 3. Computability and computational complexity of analog circuits

4. Biochemical compiler

5. Examples of function/program compilation into biochemical reactions 6. Conclusion and perspectives

MAPK Input/Output Function

Dose-response diagrams alias Bifurcation diagrams

biocham: load(library:examples/mapk/mapk).

biocham: dose_response(‘E1',1.0e-6,1e-4,200,‘PP_K').

MAPK has (at least) the function of an analog/digital converter in the cell.

How to program ^!"

# % !^" with reactions ?

Hill function ^!"

# % !^"

[Huang Ferrel 96 PNAS]

n ≈ 4.9 at 3^rd level n ≈ 1.7 at 2^nd level

n = 1 at 1^st level (Michaelis-Menten)

How to Program a function f ^{: 𝑅 → 𝑅} with Reactions ?

First idea: GPAC f(t) General Purpose Analog Computer [Shannon 41]

Inputs: t, x, y, z, … Circuit built from 1. Constant unit 2. Sum unit

3. Product unit

4. Integral ∫ 𝑥 𝑑𝑦 unit

𝑥 + 𝑦 ^/.!.1 𝑥 + 𝑦 + 𝑧 z ^/.3 _↓

dz/dt=k(xy-z)

=0 when z=xy

x → 𝑥 + 𝑧^!

dz/dt=x z= ∫ 𝑥 𝑑𝑡 x ^/.! 𝑥 + 𝑧

𝑦 ^/.1 𝑦 + 𝑧 𝑧 ^/.3 _↓

dz/dt=k(x+y-z)

=0 when z=x+y

GPAC Design of the Cosine Oscillator

Cosine(t) is the solution of the ODE y’’(t) = -y(t) with y(0)=1 1. GPAC circuit:

2. Rewriting in pseudo reactions

biocham: present(y,1).

biocham: compile_wgpac(y::integral integral-1*y).

present(x2,-1).

fast*[x2]*[y]for _=[x2+y]=>x1.

fast*[x1]for x1=>_.

_=[x1]=>x0.

_=[x0]=>y.

• Pb negative concentration values x2=-1

• Pb GPAC circuits with no solution or several

• The errors for + and * accumulate along the circuit May not scale up.

• Better design ?

Computable Analysis

Church-Turing thesis: single notion of computability, universal Turing machines Definition. A real number r is computable if there exists a Turing machine with Input: precision pÎN

Output: rational number qÎQ with | r-q |<2^-p

Example. Rational numbers, limits of computable Cauchy sequences

π, e

machine that computes f(x) with an oracle for x.

Example. Polynomials, trigonometric functions, …

Counter-example.

⌈x⌉

GPAC-generable & GPAC-computable Functions

Definitions by Polynomial Initial Value Problems (PIVP) instead of GPAC circuits:

Definition. [Graça Costa 03 complexity] A real function f:R®R is PIVP-generable If there exist a vector of polynomials pÎRⁿ[Rⁿ] and of initial values y(0) and a function y:Rⁿ®R such that y’(t)=p(y(t)) and f(t)=y₁(t)

Example. Oscillator cos(t)

Definition. [Graça Costa 03 complexity]A real function f:R®R is PIVP-computable if there exists vectors of polynomials pÎRⁿ[Rⁿ] and qÎRⁿ[R] and

a function y:Rⁿ®R such that y’(t)=p(y(t)) , y(0)=q(x) and lim

R→S𝑦_T(𝑡) = 𝑓(𝑥) Example. MAPK function PP_K(RAS)

Theorem. [Bournez Campagnolo Graça Hainry 06]

A real function is computable iff it is PIVP-computable.

Theorem. [Pouly 15 PhD thesis] A real function is computable in Ptime iff it is PIVP-computable with a trajectory of polynomial length.

Biochemically Computable Functions 1/2

Theorem. (positive systems) Any PIVP-computable function can be encoded by a PIVP on R⁺ of double dimension, preserving polynomial time complexity.

Proof idea. Encode y_iÎR by y^-_i y⁺_iÎR⁺ such that y_i= y⁺_i- y^-_i at each time Let p_i(y⁺₁, y^-₁,…, y⁺_n, y^-_n) = p_i[y = y⁺_i- y^-_i] and p_i = p⁺_i- p^-_i

y⁺_i‘ = p⁺_i - f_i y⁺_iy^-_i y⁺_i(0) = max(0, y_i(0)) y^-_i‘ = p^-_i - f_i y⁺_iy^-_i y^-_i(0) = max(0, -y_i(0))

Where f_i are positive coefficient polynomials such that f_i > max(q⁺_i, q^-_i ) Implementation by biochemical reactions ?

• Fast annihilation reactions: y⁺_i +y⁻_i f_i

→ _↓

• n-ary catalytic synthesis reactions for each monomial m⁺_i,jin p⁺_i, m⁻_i,jin p⁻_i:

M_{i, j} ⁺ m⁺_i,j

y⁺_i + M_{i, j}⁺ M_{i, j} ⁻ m⁻_i,j

y⁺_i+ Mi_{, j}⁻

Biochemically Computable Functions 2/2

Theorem. (binary reactions) [Carothers Parker Sochacki Warne 2005]

Any PIVP can be encoded by a PIVP of degree £ 2.

Proof idea. Introduce variable v_i1,…,in for each monomial y₁ⁱ¹…y_nⁱⁿ We have y₁=v_1,0…,0, y₂=v_0,1,0…,0 ,…

y’_i is of degree one in v_i1,…,in

𝑣^]_{𝑖1,… , 𝑖𝑛} = ∑^b_/cd𝑖_𝑘 𝑣𝑖1,… , 𝑖𝑘 − 1, … , 𝑖𝑛 𝑦^]_𝑘 is of degree at most 2.

(Trade high dimension for low degrees…)

Theorem. Elementary reaction systems on finite sets of molecules (no polymerization) are Turing-complete under the differential semantics

Reaction systems are not Turing-complete in the discrete semantics (Petri Nets) unless adding unbounded complexation (polymerization) [Cardelli Zavattaro 2008]

or unbounded membranes [Busi Gorrieri 2006, Paun 2000, Berry Boudol 92 ]

Biochemical Oscillator/Function Cosine

biocham: compile_from_expression(cos,time,cos(t)).

present(cos(t)_p,1).

_=[z_p_2]=>cos(t)_p.

_=[z_m_2]=>cos(t)_m.

_=[cos(t)_m]=>z_p_2.

_=[cos(t)_p]=>z_m_2.

biocham: present(x_p, 4).

biocham: compile_from_expression(cos,x,cos(x)).

present(cos(x)_p, 1).

_=[g_m]=>g_p. _=[g_m+cos(x)_m]=>z_p_4.

_=[x_p]=>g_p. _=[g_p+cos(x)_p]=>z_p_4.

_=[g_p]=>g_m. _=[x_p+cos(x)_m]=>z_p_4.

_=[x_m]=>g_m. _=[x_m+cos(x)_p]=>z_p_4.

_=[g_m+z_p_4]=>cos(x)_p. _=[g_m+cos(x)_p]=>z_m_4.

_=[g_p+z_m_4]=>cos(x)_p. _=[g_p+cos(x)_m]=>z_m_4.

_=[x_m+z_m_4]=>cos(x)_p. _=[x_m+cos(x)_m]=>z_m_4.

_=[x_p+z_p_4]=>cos(x)_p. _=[x_p+cos(x)_p]=>z_m_4.

_=[g_m+z_m_4]=>cos(x)_m. _=[x_p+cos(x)_p]=>z_m_4.

_=[g_p+z_p_4]=>cos(x)_m. _=[x_m+z_p_4]=>cos(x)_m.

Compilation of MAPK Sigmoid Function _#

^! _%!

?

• Manual solution :

Elegant but exponential time complexity on x=0 (exponential amplitude on auxiliary component y₁)

• Generic compiler-generated solution:

biocham: compile_from_expression(id*id/(1+id*id),time,sigmoid).

59 reactions on 20 species Polynomial time

yet one component diverges

• MAPK: 30 reactions, no divergence

Logical Preconditions on Reactions

[Senum Riedel 2010]

Conjunction

Disjunction

Negation better implementation

with absence indicator _{# % !}^# _e

Implementation of Sequentiality and

Program Control Flow by Absence Indicator

[Huang Jiang Huang Cheng 2012 ICCAD]

[Huang Huang Chiang Jiang Fages 2013 IWBDA]

Specification and Compilation of… Life

while true {growing; replication; verification; mitosis}

Compilation of sequentiality and loop with program control variables 50 reactions

Cyclin D, Cyclin E, Cyclin A, Cyclin B [Paul Nurse 2001]

Cyclins as necessary markers for implementing sequentiality

Enzymatic Implementation in non-living Protocells

Formal synthesis/degradation reactions as enzymatic activation reaction DNA-free microfluidic vesicles

[Franck Molina, CNRS Sys2diag]

Differential Diagnosis of Diabete

Conclusion

• Binary reaction systems over a finite set of molecules (without polymerization) are Turing-complete under the differential semantics

– PIVP definition of computable function

– Notion of computational complexity as trajectory length of stabilizing PIVPs

• Generic biochemical compiler of i/o functions

– Input: Function specification by PIVP, mixed digital-analog program – Output: system of binary reactions with mass action law kinetics – Exact characterization of the result for an ideal fluid implementation

• The natural MAPK program has a better complexity than our generated code

– Comparison to natural circuits (BioModels repository) – Code optimization ? Refactoring ? Alternative schemes ?

• Formal ground for modeling evolution: Circuit ⟷ Function

– Probably Approximatively Correct [Valliant 2013] ↻

mutation

Conclusion of the Course on some Hot Research Topics

and Thesis Subjects

1. Compilation of controllers and programs in biochemical reactions

– Design methods for digital and analog systems

– Microfluidic implementation of artificial reaction programs in protocells

2. Machine learning models from traces

– Probably Approximately Correct learning – Artificial evolution of reaction systems

3. Feedback loops in reaction systems

– Thomas’ rules for multi-stationarity in reaction systems – Implementation of Soliman’s theorem

4. Model reduction

– Tropical algebra methods for identifying slow/fast regimes – Symbolic computation

5. Multi-scale stochastic and deterministic models

– Hybrid stochastic-deterministic simulation – Model reduction