1. University of Wisconsin – Madison USA
A topological view of biological computation and new invariants to study their stability
Amir Assadi
University of Wisconsin – Madison USA
BIT June 30, 2018 Beijing

2. Thanks to Collaborators
Huijing
Gao
Mitchell
P. Thayer
Yinzhe
Gao
Yuan
Wang
Kaixi
Zhu

3. Thanks to BIT Colleagues & Students

4. Observations from Topology
In Topology, we study structures and other objects or phenomena “collectively”
Category theory – objects are smooth manifolds, vector bundles…
Statistical manifolds – points are probability distributions
Moduli spaces – points are mathematical structures
Construction of auxiliary spaces and algebraic structures to “quantify” or represent qualitatively invariants of structures.
Examples – Euler characteristic, moduli of Riemann surfaces, homology groups, classifying spaces for bundles, etc…

5. Observations from Dynamical Systems
Ex 1) state of dynamic brain activity
Example 2) state of biological communication via auto- inducer leading to biofilm formation
Example 3) state of brain, activities measured fMRI data
Example 4) state of molecules in Atmospheric Chemistry
Example …. States of the system are vectors in an n-dim space

6. Examples of Computation in Biological Systems
Brain Networks as Dynamical Systems

7. Examples of Computation in Biological Systems Formation of Bacterial Biofilms as Complex Dynamical Systems

8. Brain Signal Processing
Multichannel Observations Measure States of a Complex Dynamical System Z = multi-dimensional signal
Repeated Experiments yield “phenotypic variations”
Model States of Dynamics of Z as a Curve K
Repeated Experiments (more data sets)  …
Samples from Space of Signals

9. Sketch of Idea
Find “Complex Dynamics Invariants” (CDI) of sample curves K
Construct a Topological Space G for the CDI of Z (and/or other examples of Z)
Model Statistical Topology on G
Certain Dynamical Systems provide models for “computation”

10. Examples of Applications
Quantitative Models for Phenotypic Variation
Genomic Sources for Phenotypic Variation
Topological Invariants for Variation of Dynamics in Brain Activity
Modeling Evolution in biology TOGETHER with all phenotypic variations
Topology provides a Paradigm Shift for Modeling Phenotypic Variation.

11. State Space of a Complex Dynamical System
Ex 1) state of dynamic brain activity
Example 2) state of biological communication via auto- inducer leading to biofilm formation
Example 3) state of brain, activities measured fMRI data
Example 4) state of molecules in Atmospheric Chemistry
Example …. States of the system are vectors in an n-dim space

12. Types of measurements in Atm. Chem`
X (~Lon)
Y (~Lat)
Z (~Alt)
Temoral
Ground stations
stationary
long term
(months? years?)
Weather balloon
“Radiosonde”
X ~ X(0)
Y ~ Y(0)
0 → 30 km
2 hours,
periodic
Research flights
varies
~ hours
Satellites
Regional or global
… it’s complicated
Varies from daily to weekly

13. Space of Observable Variations of States of a Complex Dynamical System X = Quasi-Steady State Space of X
Trajectories in Quasi-Steady State Space of X (QS-SS) provide models of dynamic processes that occur and we observe
For example, various possible ways the a group of bacteria G does quorum sensing, and ends up with a biofilm.
Formation of the biofilm is the end-result of the trajectory of states of the complex dynamical system represented by G
The N observation channels that measure the states of G provide N observable coordinates, though the complex dynamical system G could have more observable parameters that vary in the course of quorum sensing.

14. Space of Observable Variations of States of a Complex Dynamical System X = Quasi-Steady State Space of X
Trajectories in Quasi-Steady State Space of X (QS-SS) is a curve in N- dimensional space
The vectors of times series given by observation channels provide an algebraic model of the “biological computation” as observed by N sensors
There are many “variations” in the course of events in quorum sensing that results in biofilm formation.
Repetition of experiments result in variation in dynamic activities of the brain

15. Topological Features of Trajectories
Trajectories as Curves in n-dimensional space
Geometry of piecewise smooth curves in n-space
Velocity (tangent) and Normal vectors span a 2-plane W in n-space
W is an element of the Grassmannian of 2-dim subspaces in n-space
Smooth variation of W(s) yields a smooth curve C in the Grassmannian G(2,n)
What about non-smooth variations?
Here we are interested in “Critical Events” that are Discontinuities
Topologically Speaking, Primary Invariants are connected components of C
Higher Order Invariants of C (critical points of C = critical events in dynamics)

16. Topological Features of Trajectories
Trajectories as Curves in n-dimensional space
Geometry of piecewise smooth curves in n-space
Grassmannian of 2-planes in n-space
Grassmannian of 3-dim subspaces in n-space
Statistics of Grassmann Invariants = Statistics of phenotypic variation
Distribution of Grassmann invariants given by samples from phenotypic variation

17. Application to Theoretical Biology
Modeling Phenotypic Variation via Path-space of a topological state space
Phenotypic Variation as variation of trajectories of biological dynamics
Algebraic Geometric Description of Biological Dynamics
Biological Communication as precursor for Biological Computation
Algebraic Geometric Description of Biological Computation
Modeling Evolution in biology as evolution of Biological Computation
Trajectories of evolution in the Space of all “biological computations”

25. Grassmann Invariants for a Trajectory
the Grassmannian Gr(1, V) is the space of lines through the origin in V = the projective space of one dimension lower than V.
Gr(2, V) parametrizes all 2-dimensional linear subspaces of the vector space V.
Gr(3, V) parametrizes all 3-dimensional subspaces
Grassmannians are compact smooth manifolds. In general,
smooth algebraic variety, of dimension r(n-r). Plücker coordinates….
Grassmannian Gr(r, V)

26. G(r,n) = O(n)/O(r)xO(n-r) …. Oriented G(r, n) = SO(n)/SO(r)xSO(n-r) …
Gr(r, n) also denote the Grassmannian of r-dimensional subspaces of n-space
Each G(r, n) has a canonical vector bundle that generalizes the notion of the canonical line bundle over the projective space G(1, n).
G(r, n) approximates the (universal) classifying space for r-dimensional bundles
G(r , n) is related to G(r-1, n) by a smooth fiber bundle. For examples, G(3, n) is a fiber bundle over G(2, n).
Oriented 2-planes (G(r, n) for n-planes) provide a double cover of the un-oriented Grassmannian.
G(r,n) = O(n)/O(r)xO(n-r) …. Oriented G(r, n) = SO(n)/SO(r)xSO(n-r) …
Grassmannian Gr(r, V)

27. Multi-channel Observables from Complex Dynamical System
Example 1) EEG channels C1, C2, … Cn measure brain activity
Example 2) measurement of auto-inducer and other biomolecules in Quorum Sensing, leading to biofilm formation
Example 3) Intensity of voxels in fMRI data from brain
Example 4) Measurement of molecules in Atmospheric Chemistry
Example ….

28. State Space of a Complex Dynamical System
Ex 1) state of dynamic brain activity
Example 2) state of biological communication via auto- inducer leading to biofilm formation
Example 3) state of brain, activities measured fMRI data
Example 4) state of molecules in Atmospheric Chemistry
Example …. States of the system are vectors in an n-dim space

29. Space of Observable Variations of States of a Complex Dynamical System X = Quasi-Steady State Space of X
Trajectories in Quasi-Steady State Space of X (QS-SS) provide models of dynamic processes that occur and we observe
For example, various possible ways the a group of bacteria G does quorum sensing, and ends up with a biofilm.
Formation of the biofilm is the end-result of the trajectory of states of the complex dynamical system represented by G
The N observation channels that measure the states of G provide N observable coordinates, though the complex dynamical system G could have more observable parameters that vary in the course of quorum sensing.

30. Space of Observable Variations of States of a Complex Dynamical System X = Quasi-Steady State Space of X
Trajectories in Quasi-Steady State Space of X (QS-SS) is a curve in N- dimensional space
The vectors of times series given by observation channels provide an algebraic model of the “biological computation” as observed by N sensors
There are many “variations” in the course of events in quorum sensing that results in biofilm formation.
Repetition of experiments result in variation in dynamic activities of the brain

31. Intelligent Behavior in Bacteria
Biological Computation at Molecular Scales
Intelligent Behavior in Bacteria

32. Example: Biofilms
Formation of bacterial biofilms (quorum sensing) is a highly complex dynamical behavior
Studied for decades by microbiologists and molecular biologists
Key role in bacterial toxicity, pathology as well as bioremediation.
Cystic Fibrosis (P. aeroginosa), infections and diseases of animal airways, urinary tracts …

34. Biofilms
Advances in bacterial genomics reveal that Quorum Sensing involves a highly organized system of biochemical interactions that have genomic origin, reflect evolutionary and physiological diversification
Some researchers refer to it as “the language of bacteria.”
Potentially, QS provides key information regarding therapeutic strategies for defending against Cystic Fibrosis and other bacterial infections.
Genetic engineering is considered to use QS research for genomic design and bacterial biofilm formation in environmental bioremediation.

35. Steps in biological communication
Evolution of communication:
Biochemical interactions
Exchange of specialized biomolecules
(Autoinducers AI1, AI2 … )
Intercellular exchange of autoinducers adapt and evolve …
Messages from one genome to another genome!
Outcome:
Biological communication

36. Steps in Biological Computation
How do they compute?
Quorum Sensing
Chemotaxis
Directed Movement
Cooperative Behavior
Formation of Biofilms …

37. What do we mean by biological computation
Input-Output Model for a biological system.
“Biological Data” is probabilistic/statistical in nature: We sample probability distributions of events/measurements.
Output responses are also statistical, even for “the same input.”
Biological systems are dynamic, non-conservative, open, and their underlying processes vary with time, depend on energy, and many environmental factors.

38. Biological Computation
Biological computation is a metaphor for:
(a) probabilistic rules that govern the input-output behavior of a given complex biological system;
(b) these rules could be understood in quantitative terms, and could be implemented by some “possibly theoretical digital” or an analog device for repeated experiments;
(c) the computational models in (b) can be validated by means of observed behavior of the original biological system.

39. Why Biological Communication?
BC is distinguished from general-purpose biochemical exchanges among organisms by virtue of the omni- presence of genomic source and target of its patterns of exchange.
Biological communication is the result of complex evolutionary events that are fundamental in emergence of intelligence, multi-cellularity, and biological mechanisms of information processing.

40. Biological Communication
Biological communication could be among systems that need not be cells. A special form of BC is intercellular signaling.
QS is an example of biological communication within members of a species (and occasionally among different species.)

41. Biological Communication…. II
It is plausible (but far from proven!) that patterns of observations of QS in a species (e.g. R.sph.) fall into a finite and orderly collection of DYNAMIC EVENTS.
Static observations (single-events with no mention of time as a parameter) need not satisfy such regularity restriction.

42. Biological Communication…. II
Intercellular signaling is the result of complex evolutionary events that are fundamental in formation of patterns of information processing.
Mathematical modeling of biological mechanisms of information processing among biosystems will greatly contribute to:
engineering of dynamic molecular networks in cells
understanding increasing complexity of multi-cellular organisms
Evolution of complexity and complex behavior

43. Biological Communication The Physics
Collect observations from the dynamical states of intelligent systems, e.g. bacteria engaged in biological communication
We also observe regularity or irregularity of behavior in the biosystem, e.g. a self-organizing system of bacteria
On finer scales, we observe traffic of messages, for example, autoinducers (AI) in quorum sensing, …
These observations could be encoded as a collection of vectors (arrays of features), call it S in a vector space V where N = dim(V)
When observations are dynamic, data vectors have additionally a time-series structure.

44. Biological Communication The Mathematical Problem
When does S embody a process (set of features) that we could call a model for (intercellular) biological communication?
To answer this question for QS,
we need algorithms that extract dynamic patterns from collections of genomic measurements that underlie quorum sensing
We need a theory that shows when the ensemble of patterns could be “approximated by mathematical quantities ” (we called them S)
Design a model that identifies the logical relationships of the patterns as a combination of Boolean and polynomial operations.

45. Biological Communication The Algebra
A certain class of computations could be described by polynomial rules:
the inputs are arrays of numbers (a1, a2, …, am)
we substitute (a1, a2, …, am) for variables: (x1, x2, …, xm) = X
variables appear in a collection of polynomial rules: f1(X)= Y1, f2(X)= Y2, …, fm(X)= Ym.
Equivalently Y=(Y1, Y2, …Yn), F(X, t)=Y(t).

46. Biological Communication The Geometry
Let G be the graph of the function F.
It is the zero set of the system of equations
F(X , t) - Y(t) = 0
G is called an algebraic set or variety, the main object of study in algebraic geometry.
Addition of time parameter t is needed for dynamics.)
To solve the (BC) problem for QS, we construct a representation of the genomic (biomolecular) data S that could be realized as a geometric model for G.
This is equivalent to extracting the polynomial rules
f1(X, t), f2(X, t), …, fm(X, t).

47. Biological Communication The Technicalities
Polynomial equations over real numbers could have an empty set of solutions!
Over complex numbers, the Fundamental Theorem of Algebra assures us that any (non-constant!) polynomial always has zeros.
Dynamic events provide data (time-series) whose Fourier Transform is naturally defined over complex numbers!
Therefore, one way to approach the (BC) Problem is to collect dynamical data and take advantage of Fourier Transform.
Algebraic geometry over complex numbers is needed to construct G and to extract the rule Y=F(X).

48. Biological Communication Conclusion
Collecting data from QS needs biological expertise for molecular biology of the bacteria and research in their systems biology (hence networks). It requires design of novel biological experiments!
Since biological data has noise, we must use probabilistic arguments (Bayesian networks organize the combinatorics of conditional probabilities!) We also use adaptive methods of estimation.
Innovations? Bringing together all these diverse areas, elucidation and the use of biological communication as genomic events underlying information processing, and extracting polynomial rules from noisy data!