# FY2010 Mathematical Biology Unit

# Mathematical Biology Unit

**Principal Investigator: Robert SinclairResearch Theme: Interactions between Mathematics and Biology**

### Abstract

This has been a year of consolidation and expansion in our research. We are now tackling problems of direct relevance to neuroscience, the evolution of development, and the deep evolutionary history of bacteria and archaea. The techniques we are using range from classical mathematical proofs to computational analyses of genome data.### 1. Staff

- Dr. Gunnar Wilken, Researcher
- Ms. Shino Fibbs, Research Administrator

### 2. Collaborations

**Theme: Geometry and visualization of integrable systems and non-integrable systems**- Type of collaboration: Joint research (JSPS Challenging Exploratory Research Grant Number 22654010)
- Researchers:
- Professor Martin Guest, Tokyo Metropolitan University
- Dr. Takashi Sakai, Tokyo Metropolitan University
- Emeritus Prof. Richard Palais, Brandeis University

### 3. Activities and Findings

#### 3.1 Evidence for Exposure to Spliceosomal Activity in Prokaryotic Coding Sequence

One of the major differences between Prokaryotes (bacteria are a prominent example) and Eukaryotes (including all animals and plants) is the existence of what one might call an editing system (the spliceosome) in the molecular information processing machinery of Eukaryotic cells. This editing system allows our cells to interpret their genes in different ways in different bodily organs, for example, and is important for the generation of the organismal complexity of almost all visible forms of life. The fact that Prokaryotes do not have this system has been a source of debate for decades. Did Prokaryotes never have this system, or did they lose it as a result of some sort of genomic streamlining? The focus of our work has been to look for a particular *asymmetry* in coding DNA which one would only expect to see in organisms which have, or have had, a spliceosome. The major result of this work has been that we do see this asymmetry in Prokaryotic genomes. If independently confirmed, this result would help us to understand the differences between ourselves and many of our pathogens. Thus, the work has potential applications outside of discussions of the early origin of life on Earth. A draft manuscript describing the first results of this work has been posted on Nature Precedings:

http://precedings.nature.com/documents/5770/version/1

#### 3.2 The Concept of Complexity and its Relationship to Modularity in Evolutionary Theory

It is worth thinking about what complexity might mean in the context of evolutionary theory, from a theoretical point of view. First, concepts need to be richly related to each other in order to be truly powerful – there can and should be an *ecology of theory*. The aim of this work is to elucidate a possible relationship between the abstract concepts of complexity and modularity. We have begun using boolean networks with inhibitory connections as our abstract model organism. Concretely, computer proofs suggest that selection for complexity can induce modularity. A fundamental question here is what selection for complexity might mean. Adaptive immune systems provide the most immediate examples, and they are all modularly constructed.

We conjecture that a true understanding of evolution by theoretical means will require a number of such mathematical model organisms, each illuminating a different aspect of the whole. We have been performing intensive computations which constitute the computer proofs. The computations are of a graph-theoretical nature. In future work, we intend to introduce new abstract model organisms. These would involve probabilistic and game-theoretic aspects.

#### 3.3 The Geometry of Dendritic Branching

This is joint work with Drs. Y. Kim and E. De Shutter (PI) of the Computational Neuroscience Unit. The neurons of our nervous system possess often quite complex tree-like structures, called dendritic trees. These are sometimes almost perfectly planar, but typically fill some volume. Their function is, broadly speaking, to receive input to the neuron. Dendritic trees are themselves also capable of some information processing. Their properties, including purely geometrical ones, are therefore of great interest to neuroscience. A single dendritic tree is, generically, a rooted binary tree. It was observed several decades ago that the local branchings of dendritic trees are typically flat. This raises a number of fundamental questions such as: (i) What is the most appropriate way to measure "flatness"? (ii) What would be the distribution of this quantity for "random binary trees"? (iii) What can we potentially learn about the growth of neurons from comparing the distribution of this quantity in "random binary trees" to its distribution in actual dendritic trees? (iv) Finally, what do we mean when we say "random binary tree"? This work involves stochastic geometry and generalizations of Steiner tree theory.

#### 3.4 The Fermi-Pasta-Ulam Experiment

This is joint work with Prof. M. Guest and Dr. T. Sakai of Tokyo Metropolitan University, and Emeritus Professor Richard Palais of Brandeis University. In the very earliest days of scientific computing, the famous physicist Enrico Fermi initiated a computational study of the thermal relaxation of materials (lattices) with nonlinear interactions. This was perhaps the first scientific computation ever performed (in 1953), and has had a lasting impact on the theories of chaos and solitons. The reason for its impact was that its results were entirely unexpected. Instead of becoming thermalized (reaching thermal equilibrium), the lattice seemed to be locked into some sort of periodic behaviour. Since this time, mathematicians have struggled to determine the deeper reasons for this observed behaviour. In our work, we are currently developing software with two independent goals: (i) To understand the original computation (this is not a simple matter because Fermi died before the work was complete, so it was never formally published), and (ii) to enable further mathematical study of the problem, considering a wider variety of models and interactions than the original experiments did.

We are also actively investigating connections between the dynamics of such lattice systems and various biological phenomena, such as peristalsis and collective neuronal oscillations.

#### 3.5 Mathematical Analysis of Models of Integrated Pest Management on two Patches

This has been joint work with Dr. P. Georgescu of the Technical University of Iaşi. Integrated pest management is an environmentally friendly way to control pests, often using their natural enemies as the central control mechanism. Mathematical models are difficult to extend to more than one ecological patch, but we have taken a step in this direction. We studied a two-patch model with one pest and a horizontally transmitted viral infection. A numerical investigation of the phase space of the model led to the demonstration that increased frequency of pesticide use may not provide pest control in this model. Strong theoretical results were obtained for this ecologically motivated case of two patches, using Floquet theory for periodic systems of ordinary differential equations.

#### 3.6 Transfer of Long-Term Memory

This project is located in neuroscience and has partly been joint with Dr. Nozomu H. Nakamura from the Naito Unit. We have investigated how episodic memory content, which is widely agreed to be dependent on the hippocampus, gradually transforms into remote (semantic) memory and thus becomes independent of the hippocampus. We investigated fundamental differences in the neuronal networks supporting the respective forms of memory and proposed abstract network paradigms related with them. The project was presented as a poster at SfN 2010 in San Diego, USA, and has been discussed extensively with Prof. Kaori Takehara-Nishiuchi at the University of Toronto, Canada.

#### 3.7 Complexity of Higher Order Rewrite Systems

This project is joint work with Prof. Andreas Weiermann at the University of Ghent, Belgium. Lindenmayer systems, which have numerous applications in biological modeling, form a subfamily of rewrite systems whose theory is part of mathematical logic and computer science. A prototype of expressive higher order rewrite systems is Gödel’s T formulated in typed lambda-calculus. A detailed analysis of its complexity was published in Lecture Notes in Computer Science (LNCS) in 2009 (with Weiermann). Inspired by substantial discussions with Prof. William A. Howard we could improve the methodological approach, simplify the proofs, and are now finalizing a full version in extensively rewritten form for publication in Logical Methods in Computer Science. The work has been presented at a workshop on proof theory and computability theory in Sendai, organized by Tohoku University and JAIST. An article on ordinal arithmetic (with Weiermann) is now available online via early view service of Mathematical Logic Quarterly: http://onlinelibrary.wiley.com/doi/10.1002/malq.200910125/abstract

#### 3.8 Elementary Patterns of Resemblance

This project has its origin in the area of proof theory and is joint work with Prof. Timothy J. Carlson at the Ohio State University, USA. Patterns of resemblance (Carlson, 2001) provide a new approach to ordinal notation systems which in turn open the road for proofs of consistency of mathematical theories and the measurement and hence comparison of their strengths. Patterns are finite structures consisting of nested trees evolving along complex sequences of intriguing self-resembling repetition. An article on normal forms of patterns (with Carlson) has recently been accepted for publication in the Journal of Symbolic Logic, an article on pure patterns of order two (with Carlson) is currently under review for publication in Annals of Pure and Applied Logic. The research on patterns of resemblance was presented at the Satellite Event for Logic and Set Theory of the ICM 2010 in India as an "Accepted Paper".

### 4. Publications

#### 4.1 Journals

- Georgescu, P., Dimitriu, G., Sinclair, R. Impulsive Control of an Integrated Pest Management Model with Dispersal between Patches.
*Journal of Biological Systems***18**, 535-569, doi:10.1142/S0218339010003561 (2010).

#### 4.2 Books and other one-time publications

Nothing to report#### 4.3 Oral and Poster Presentations

- Sinclair, R.
*A Quantative Approach to Investigating the Hypothesis of Prokaryotic Intron Loss*, SMBE 2010, Lyon, France, 07 04-08, 2010 - Sinclair, R.
*What causes Flat Dendritic Bifurcations?*, Global COE Seminar, Kyoto University, Apr 23, 2010 - Sinclair, R.
*Thoughts on complexity's place in evolutionary theory*, OIST Winter Course "Evolution of Complex Systems", Okinawa Institute of Science and Technology, Dec 8, 2010 - Sinclair, R.
*Natural Computation*, JAVA Tutorial Seminar, Tokyo Metropolitan University, Tokyo, Dec 13, 2011 - Wilken, G.
*Tracking Chains of Sigma_2-Elementarity*, Conference on Mathematical Logic and Set Theory; Satellite Conference of the ICM 2010, Chennai Mathematical Institute, Chennai, India, Aug 15, 2010 - Wilken, G., Nakamura, N.
*Memory Transfer from Hippocampus to Neocortex*, The 40th Annual Meeting of the Society for Neuroscience, San Diego, USA, Nov 15, 2010 - Wilken, G.
*Derivation lengths Classification for Goedel's T revisited*, Workshop on Proof Theory and Computability Theory, Sendai, Miyagi Pref., Feb 21-24, 2011

### 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report### 6. Meetings and Events

#### 6.1 Research Visit

- Date: February 20-23, 2011
- Venue: Onna-son Campus Lab 1
- Visitors: Prof. Martin Guest and Dr. Takashi Sakai of Tokyo Metropolitan University
- Purpose: To continue our mathematical visualization and simulation discussions and also to extend our JAVA programming work. This visit was in the context of the project "Geometry and visualization of integrable systems and non-integrable systems" (JSPS Challenging Exploratory Research Grant number 22654010).

#### 6.2 Research Visit

- Date: March 8-11, 2011
- Venue: Onna-son Campus Lab 1
- Visitor: Asst. Prof. Kaori Takehara-Nishiuchi of the University of Toronto
- Purpose: To discuss theoretical approaches to understanding the phenomenon of memory consolidation.

#### 6.3 Seminar

- Title: "On Toponogov's comparison theorem"
- Date: December 22, 2010
- Venue: Onna-son Campus Lab 1
- Speaker: Dr. Kei Kondo of Tokai University

#### 6.4 Seminar

- Title: "Physiology of Frontal-temporal Lobe Memory Network"
- Date: March 9, 2011
- Venue: Campus Seminar Room C210
- Speaker: Asst. Prof. Kaori Takehara-Nishiuchi of the University of Toronto

### 7. Others

Dr. Sinclair has been involved in a number of outreach activities in Okinawa throughout the year.

- "Looking for new mathematics in the living world", presented as a part of the OIST Onna Campus Sunday, November 28 (2010). Viral infections were described from a computational point of view, using audience participation.
- Problem Solving, Reasoning & Proof, Communication, Connections, and Representation", the keynote presentation for the 2011 DoDEA Pacific Far East MathematicaFest, February 7 (2011). This was a discussion of the nature of Mathematics, using the Fibonacci sequence as an example.
- "Mathematics and the Natural Sciences", a talk to the winners of the OIST Essay Contest, at the Sun Sea Science and Students Workshop, March 14-18 (2011). Mathematics is not a study of any one thing. Rather, it is a study of relationships and patterns. It is the very fact, that mathematical knowledge is not tied to the properties of any specific thing or system, which gives it its power of generality. Mathematicians working with natural scientists have a special responsibility to ensure that the chosen mathematical approach actually does make sense, both mathematically and also in the scientific context. Finally, mathematicians need to be open to the fact that natural sciences often motivate new fields of mathematics, so mathematics has as much to benefit from science as science from mathematics. These issues were discussed using concrete examples.