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2007

Volume 49, Issue 1, pp. 1-176

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Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 49, pp. 1-1 (1 page) | Cited 2 times

Online Publication Date: January 31, 2007

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In the November 10, 2006 issue of Science, which was devoted to the “Glorious Sea Urchin,” scientists report the similarities between the common purple sea urchin genome and that of the human. This similarity is of course no accident—we all come from the same organism. Unraveling the connections between living organisms through the study of their genomes is phylogenomics, which combines evolutionary studies and genomics in an integrated approach. This issue’s Survey and Review article by Lior Pachter and Bernd Sturmfels is an invitation to the mathematics of phylogenomics.
The paper begins with a lucid and lively introduction to the basics of genomics. The goal of phylogenomics is to identify common ancestry between organisms and to understand their evolution by analysis of their genomes. This is done by a process called alignment, which identifies highly conserved sequences. Such a sequence would appear across a wide range of organisms and points to the likelihood that these organisms share a common evolutionary ancestor. The authors state an audacious “Meaning of Life” conjecture, which maintains that a certain 42‐letter sequence is present in the genome of all vertebrates.
Hidden Markov models are a key tool in genomics. What is particularly exciting about the present work is how the authors use algebraic methods as a framework for their investigation, and how these methods point to practical algorithms for gene finding, sequence alignment, and evolutionary modeling. The article concludes with a compelling example calculation that suggests the “Meaning of Life” sequence plays a crucial role in the biology of vertebrates.

The Mathematics of Phylogenomics

Lior Pachter and Bernd Sturmfels

SIAM Rev. 49, pp. 3-31 (29 pages) | Cited 5 times

Online Publication Date: January 30, 2007

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The grand challenges in biology today are being shaped by powerful high‐throughput technologies that have revealed the genomes of many organisms, global expression patterns of genes, and detailed information about variation within populations. We are therefore able to ask, for the first time, fundamental questions about the evolution of genomes, the structure of genes and their regulation, and the connections between genotypes and phenotypes of individuals. The answers to these questions are all predicated on progress in a variety of computational, statistical, and mathematical fields. The rapid growth in the characterization of genomes has led to the advancement of a new discipline called phylogenomics. This discipline results from the combination of two major fields in the life sciences: genomics, i.e., the study of the function and structure of genes and genomes; and molecular phylogenetics, i.e., the study of the hierarchical evolutionary relationships among organisms and their genomes. The objective of this article is to offer mathematicians a first introduction to this emerging field, and to discuss specific mathematical problems and developments arising from phylogenomics.
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Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 49, pp. 33-34 (2 pages)

Online Publication Date: January 31, 2007

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The three papers in this issue deal with how to describe spikes and plateaus; how to bound the probability that a random variable belongs to a particular set; and how to analyze thermodynamic properties of DNA and RNA.
1. If you had only a (mathematical) sound bite to explain the difference between the broad plateau of the Black Mesa in Arizona and the steep spike of the Matterhorn in Switzerland, what could you do? Thomas Hillen, in his paper “A Classification of Spikes and Plateaus,” proposes that you inspect fourth derivatives. In Flatland (a one‐dimensional domain, that is), the Black Mesa has a negative fourth derivative at its highest point, while the Matterhorn has a positive fourth derivative. For $n$‐dimensional domains it’s a little bit more complicated because one has to check whether the Hessian of the Laplacian is positive definite or negative definite. The author illustrates how the classification of plateaus and spikes benefits the qualitative analysis of several partial differential equations from mathematical physics and biology.
2. Given a random vector $X\in\real^n$ whose first two moments (the expected mean vector $\mathbf{E}X$ and the covariance matrix $\mathbf{E}XX^T$) are known, the problem considered in the second paper is to bound the probability that $X$ lies in a particular subset $C$ of $\real^n$. Here the set $C$ is defined by $m$ strict quadratic inequalities, $$C=\left\{x\in\real^n:\ x_i^TA_ix+2b_i^Tx+c_i<0, \quad 1\leq i\leq m\right\},$$ which involve symmetric matrices $A_i$, vectors $b_i$, and scalars $c_i$. The best possible (“sharp”) lower bound on the probability that $X$ lies in $C$ is an example of a generalized Chebyshev inequality.
In general, generalized Chebyshev inequalities are sharp upper or lower bounds on the probability that a random vector with given moments lies in a particular set. The first such inequality was formulated in the nineteenth century by Chebyshev and proved by his student Markov. Almost a hundred years later duality theory and optimization emerged as powerful tools for deriving Chebyshev inequalities. Since then Chebyshev inequalities have appeared in decision analysis, statistics, finance, and machine learning.
In their well‐written paper “Generalized Chebyshev Bounds via Semidefinite Programming,” Lieven Vandenberghe, Stephen Boyd, and Katherine Comanor construct two equivalent (dual) semidefinite programs for solving the above Chebyshev inequality. That is, the optimal value of these programs equals the best possible lower bound on the probability that a random vector $X$ with given mean and covariance lies in the set $C$. If you have time, savor the well‐organized and constructive proofs, and check out the two simple examples to watch the semidefinite programs in action.
3. Nucleic acid technology is based on the design of molecular systems that self‐assemble out of strands of DNA or RNA, so as to implement functions relevant to robotics, biosensing, medicine, and many other applications. Fundamental to the success of this technology is a rigorous analysis of the thermodynamic properties of nucleic acid strands. In their paper “Thermodynamic Analysis of Interacting Nucleic Acid Strands,” Robert Dirks, Justin Bois, Joseph Schaeffer, Erik Winfree, and Niles Pierce make a significant advance by devising models and algorithms, not just for a single strand, but for an entire test tube of interacting strands of nucleic acids. They do this by combining an unusual variety of different mathematical techniques.
A nucleic acid strand can be represented as a sequence of bases from a four‐letter alphabet (for DNA the bases are A, C, G, and T). Complementary bases can interact to form base pairs (for DNA these base pairs are C‐G and A‐T). The set of base pairs in a molecular conformation of interacting nucleic acid strands is called the secondary structure. In the simplest case, two fully‐complementary strands can base‐pair to each other completely, bringing to mind a ladder in which the rungs represent base pairs and the two sides of the ladder represent the backbones of the two strands. In practice, however, things are usually more complicated. Not every base is paired, and the strands might interact to form branched structures with various types of bulges and loops between the base pairs. For reasons of practicality, the authors disallow certain complicated secondary structures (pseudoknots).
An important thermodynamic property, useful for the design of nucleic acids, is the equilibrium probability of a secondary structure. It can be calculated in a straightforward way from the partition function. Calculating the partition function for a single strand requires summing the partition functions of all smaller subsequences. This can be done by a dynamic program that computes the sums recursively. By contrast, calculating the partition function for a complex of several interacting strands is much more difficult, due to the physical and mathematical issues that arise when bases from different strands pair up.
The authors present models and algorithms for computing the partition function for a complex of an arbitrary number of interacting strands. In particular, they use graph theory to map each allowed secondary structure to a unique ordering of strands (representation theorem). And they use group theory to eliminate symmetries and redundancies that would lead to overestimates of the partition function (distinguishability correction). Together, these results ensure that a dynamic program can again be used to recursively calculate the partition function.
Now the authors are ready to extend their analysis to realistic experimental conditions where an arbitrary number of strand species interact (in a dilute solution) to form a variety of complexes. The authors use combinatorial arguments to determine how many times the dynamic program must be invoked to compute the partition function for such a system. They express the computation of the equilibrium concentration of each species of complex as a convex programming problem and use the concave dual problem as the basis for an efficient numerical implementation.

A Classification of Spikes and Plateaus

Thomas Hillen

SIAM Rev. 49, pp. 35-51 (17 pages) | Cited 4 times

Online Publication Date: January 30, 2007

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In this paper we classify local maxima into spikes and plateaus. We give analytic definitions for spikes and plateaus in terms of a nonlocal gradient and a fourth order derivative. In higher dimensions the Hesse matrix of $\Delta f(x)$ is of relevance. This classification is applied to pattern formation models in mathematical physics and mathematical biology, including Cahn–Hilliard equations, chemotaxis equations, reaction‐diffusion equations, Gierer–Meinhardt models, and Gray–Scott models. We show for some of these examples that the stability of spatial patterns depends on the spike versus plateau type of the solution. We prove, for example, that scalar reaction‐diffusion equations in any spatial dimension cannot have stable spike steady states.

Generalized Chebyshev Bounds via Semidefinite Programming

Lieven Vandenberghe, Stephen Boyd, and Katherine Comanor

SIAM Rev. 49, pp. 52-64 (13 pages) | Cited 7 times

Online Publication Date: January 30, 2007

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A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result generalizes Chebyshev’s inequality for scalar random variables. Two semidefinite programming formulations are presented, with a constructive proof based on convex optimization duality and elementary linear algebra.

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Robert M. Dirks, Justin S. Bois, Joseph M. Schaeffer, Erik Winfree, and Niles A. Pierce

SIAM Rev. 49, pp. 65-88 (24 pages) | Cited 42 times

Online Publication Date: January 30, 2007

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Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single‐stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base‐pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality.
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SIGEST

SIAM Rev. 49, pp. 89-89 (1 page)

Online Publication Date: January 31, 2007

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How often is it the case that not only the applications but also the methods of analysis of an important mathematical property range all the way from the deeply mathematical to the realm of the social sciences (and beyond)? Or, for that matter, that one can learn about the property under consideration in some detail on Wikipedia? The {small world phenomenon} is an instance of all of these characteristics.
For the mathematical community, the small world phenomenon came alive with the 1998 paper of Watts and Strogatz that looked at the phenomenon from the perspective of a lattice network model and showed that the addition of a relatively small number of randomly chosen long—range connections to an otherwise locally connected lattice leads to a very good model of the small world phenomenon. Their work and others showed further that this type of model describes the actual connectivity and behavior of applications in areas as diverse as the brain of a small organism, the electrical power grid, and connectivity on the World Wide Web.
This issue's SIGEST paper, ’’A Matrix Perturbation View of the Small World Phenomenon’’ by D. J. Higham that originally was published in the { SIAM Journal on Matrix Analysis and Applications} in 2003, makes an inspired and important contribution to the mathematical understanding of the small world phenomenon by looking at it from a new perspective. As Higham discusses in the new preamble to the paper, he was motivated to formulate a model that could display the small world phenomenon and be amenable to rigorous mathematical analysis. He succeeded by considering a Markov chain that also can be thought of as a one—dimensional periodic random walk, where the standard connections to the two nearest neighbors are augmented by low probability uniform jumps to any state, also referred to as shortcuts. Higham uses matrix perturbation theory methods to study the correspondence between the probabilities of these uniform jumps and the mean hitting time (the average number of steps to get from some random state to a given fixed state) of the random walk. In particular, the mean hitting times of a Markov chain can be computed by solving a certain system of linear equations, and by use of matrix perturbation theory, Higham analyzes how these times vary with the probability of the uniform jumps. The paper shows that this model is very effective in explaining the small world phenomenon mathematically, and that in addition, its mathematical behavior corroborates behavior that was observed experimentally by Watts and Strogatz. Higham's model also has a close connection to techniques used in Internet search engines.
We hope that many SIAM readers will enjoy this interesting, accessible, and very nicely written paper that adds great insight into an important property of the natural and social worlds. And those of you with Hollywood connections will have something novel to bring up the next time the conversation turns to the ’’Six Degrees of Kevin Bacon.’’

A Matrix Perturbation View of the Small World Phenomenon

Desmond J. Higham

SIAM Rev. 49, pp. 91-108 (18 pages) | Cited 3 times

Online Publication Date: January 30, 2007

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We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440–442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean‐field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201–3204].
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Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 49, pp. 109-109 (1 page)

Online Publication Date: January 31, 2007

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Many applied mathematicians are obsessed with explaining the behavior of the world around us; we find an ability to build a model that replicates and quantifies our observations indispensable. For many of us this fascination started long ago as child’s play when we were throwing Frisbees, blowing bubbles, and—the topic of this issue’s Education section paper—toppling long strings of dominoes. Returning to these problems many years later we’ve learned to ask some more sophisticated questions: How fast do the dominoes fall? How does the speed depend upon the domino spacing? What are the relevant physical effects and nondimensional parameters?
Our paper “Domino Waves” by C. J. Efthimiou and M. D. Johnson presents a self‐contained analysis of this problem. It starts with some basic physical assumptions, then analyzes the mechanics which yields a recursive relationship between the angular velocities of successive falling dominoes, and finally, after taking a small detour to learn a few facts about elliptic integrals, determines the speed of a domino wave. The resulting prediction can easily be tested by the intrepid student. A semester of college physics and calculus should be all that a student needs to read this paper; it is a natural adjunct to a course on mechanics or mathematical modeling and could easily be developed into a lab exercise in either of these courses.
I encourage the many teachers among our readers to shake up the normal classroom routine and take a foray into inquiry‐based learning. Bring some dominoes into class (or go to YouTube.com and type in “falling dominoes”) and ask your students to build a mathematical model of what they see. After they gain a certain familiarity with the problem, give them a copy of the paper and invite them to investigate the model presented, verify the conclusions, and perhaps extend the model on their own. In this way you can start your own chain reaction of inquiry, with one gentle push.
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Domino Waves

C. J. Efthimiou and M. D. Johnson

SIAM Rev. 49, pp. 111-120 (10 pages) | Cited 1 time

Online Publication Date: January 30, 2007

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Motivated by a proposal of Daykin [Problem 71‐19∗, SIAM Rev., 13 (1971), pp. 569–570], we study the wave that propagates along an infinite chain of dominoes and find the limiting speed of the wave in an extreme case.

Book Reviews

Bob O’Malley, Section Editor

SIAM Rev. 49, pp. 123-176 (54 pages)

Online Publication Date: January 31, 2007

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This issue’s featured review covers the last two Mathematica Guidebooks (on numerics and symbolics). We owe Willy Hereman of the Colorado School of Mines much thanks for completing the huge reviewing task involved. Readers are encouraged to compare what he previously wrote about the first two volumes [SIAM Rev., 47 (2005), pp. 801–806]. Other reviews cover a wide range of topics, including much pure mathematics. The reviews, as usual, vary in style and eloquence. Some opinions are backed up by explanations or strongly worded criticisms of authors or publishers. They don’t yet often have the impact or sting of a New York Times review by the Pulitzer prize winner Michiko Kakutani, but some are equally memorable. We hope they’re also fair, pointing us toward important new literature.
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