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Course Offerings (GSAS Bulletin)

REQUIRED COURSES

Courses selected for the COB program are selected from participating home departments to provide students with a disciplinary foundation and the breadth of an interdisciplinary approach to science. All COB students, regardless of home department, are required to enroll in four semesters of the COB Research Seminar.

n addition, each student is required to enroll in required courses specific to his/her home department, as listed below. Some of these required courses, as appropriate, may also be used by students as a crossover or elective course. Additional information on these and other courses may be accessed through the Web sites of individual home departments.

Note: COB students whose home department is the Mount Sinai School of Medicine should check their graduate handbook at http://fusion.mssm.edu/ gradschool/courses/course.cfm for the description of their required biology core courses (Biochemistry and Molecular Biology: Core I and Cell Biology: Core II) and possible electives/crossover courses. All COB students may check that site for possible elective/crossover courses.

COB Research Seminar
G24.2200 Offered each term, with content varying from semester to semester. Prerequisite: enrollment in the computational biology doctoral program or permission of the instructor. 3 points per term.
The many concerted initiatives in genomics, bioinformatics, biomolecular structure determination, computational neurobiology, and biological imaging and the development of analytical and computational tools have immense ramifications on every aspect of our lives—from health to technology to law. Such developments have evolved from foundations laid by many pioneers in the biochemical sciences and allied fields. This seminar introduces students to emerging disciplines that helped establish the field of computational biology through lectures and readings from the scientific literature, both technical (journal articles) and general (books about science and scientists). It seeks to both familiarize students with the field’s evolution, as well as help students develop a critical eye for conducting research in the field.

Basic Medical Sciences (Sackler Institute)

Foundations of Cell and Molecular Biology I, II
G16.2001, 2002 Lecture and conference. I offered every fall; II offered every spring. Prerequisites: basic biochemistry and cell biology. 6 points per term.
Intensive, two-semester advanced course that provides a broad overview of nucleic acid and protein metabolism and function. The fall semester covers DNA metabolism, including DNA replication, repair, and recombination; chromatin structure; RNA transcription and processing; and translation control mechanisms. The spring semester covers various aspects of cell biology, signal transduction, and genetics. Topics include biogenesis of cellular membranes; vesicular transport; the cytoskeleton; cell differentiation and development; concepts in receptor signaling; and genetics of model organisms. Each semester consists of two or three modules that differ somewhat in organization, including the number of required lectures. Each module places significant emphasis on student-led discussions. The reading of primary research articles is heavily stressed. Grades are assigned on the basis of examination, essay, and discussion scores.

Biology

Bio Core 1: Molecules and Cells
G23.1001 4 points.
A survey of the major topics of up-to-date molecular and cellular biology, starting with molecular structure and function of proteins and polynucleic acids and ending with cell division and apoptosis.

Bio Core 3: Genes, Systems, and Evolution
G23.1002 4 points.
A survey of the major topics of modern biology, including genetics, systematics, genomics, systems biology, developmental genetics, plant biology, immunology, neurobiology, population genetics, evolution, and geobiology.

Computer Science

Honors Programming Languages
G22.3110 Prerequisite: permission of the instructor. 4 points.
In-depth examination of the four major categories of programming languages: imperative, object-oriented, functional, and logic languages. The specific languages covered include Ada, C++, LISP, ML, Prolog, and SETL. Fundamental issues of programming languages, such as type systems, scoping, concurrency, modularization, control flow, and semantics, are discussed.

Honors Analysis of Algorithms
G22.3520 Prerequisites: G22.1170 or one semester of undergraduate algorithms, and permission of the instructor. 4 points.
Design of algorithms and data structures. Review of searching, sorting, and fundamental graph algorithms. In-depth analysis of algorithmic complexity, including advanced topics on recurrence equations and NP-complete problems. Advanced topics on lower bounds, randomized algorithms, amortized algorithms, and data structure design as applied to union-find, pattern matching, polynomial arithmetic, network flow, and matching.

Mathematics

Numerical Methods I, II
G63.2010, 2020 Identical to G22.2420, 2421. Corequisite: linear algebra. 3 points.
Numerical linear algebra. Approxi-mation theory. Quadrature rules and numerical integration. Nonlinear equations and optimization. Ordinary differential equations. Elliptic equations. Iterative methods for large, sparse systems. Parabolic and hyperbolic equations.

Chemistry
Students select two courses from the following.

Biochemistry I, II
G25.1881, 1882 Identical to G23.1046, 1047. Prerequisites: V25.0243 and V25.0244, or equivalent courses in organic chemistry for G25.1881; G25.1881 for G25.1882. 4 points per term.
Two-semester course taught jointly by faculty from the Departments of Biology and Chemistry. Topics include organic and physical chemistry of proteins, lipids, carbohydrates, and nucleic acids; enzyme kinetics and mechanisms; membranes and transport; bioenergetics and intermediary metabolism; molecular genetics and regulation.

Statistical Mechanics
G25.2600 4 points.
Introduction to the fundamentals of statistical mechanics. Topics include classical mechanics in the Lagrangian and Hamiltonian formulations and its relation to classical statistical mechanics, phase space and partition functions, and the development of thermodynamics. Methods of molecular dynamics and Monte Carlo simulations are also discussed.

Biomolecular Modeling
G25.2601 Prerequisite: basic programming experience. 4 points.
Introduction to molecular modeling and simulation, including development of ab initio and semiempirical potentials, molecular mechanics, Monte Carlo simulations, and molecular dynamics simulations, both theory and practice.

Center for Neural Science

Cellular, Molecular, and Developmental Neuroscience
G80.2201 4 points.
Team-taught, intensive course. Lectures and readings cover basic biophysics and cellular, molecular, and developmental neuroscience.

Sensory and Motor Systems
G80.2202 4 points.
Team-taught intensive course. Lectures and readings concentrate on neural regulation of sensory and motor systems.

SAMPLE CROSSOVER AND ELECTIVE COURSES

Basic Medical Sciences (Sackler Institute)

Principles of Structural Biology
G16.2004 4 points.
The goal of this course is to provide students with an in-depth understanding of the structures of proteins and nucleic acids, the modes of interaction that underlie protein-protein and protein-nucleic acid recognition, and how knowledge of macromolecular structure leads to an understanding of biological processes. Topics include enzyme structure and mechanism, membrane proteins, ligand-receptor recognition, protein-protein interactions in signal transduction, molecular machines, and protein-nucleic acid recognition. The class meets three times per week—two lectures and one discussion session.

Bioinformatics
G16.2604 Prerequisites: a thorough understanding of theoretical and practical aspects of molecular biology and some university-level mathematics and statistics; no prior knowledge of computer programming or computer hardware is necessary. 4 points.
A practical course in bioinformatics that emphasizes the use of the computer as a tool for biomedical research. The course covers sequence similarity, multiple alignment, protein motifs and secondary structure, phylogenetics, genome browsers, and microarray data analysis. Students learn basic UNIX commands and write simple programs in Perl and shell scripting languages.

Advanced Topics in Structural Biology
G16.4403 Prerequisite: G16.2004. 4 points.
This course teaches students the underlying theory and techniques used in X-ray crystallography, electron microscopy, NMR spectroscopy, mass spectrometry, and computer modeling. The information in this course enables students to pursue their dissertation research in structural biology. Topics include X-ray diffraction, phasing, and refinement; cryoelectron microscopy, image processing, and tomography; multidimensional NMR spectroscopy; MALDI-TOF and Q-TOF mass spectrometry; and ab initio and homology modeling of proteins.

Fundamental Concepts of Magnetic Resonance Imaging
G16.4404 Prerequisites: calculus, linear algebra, general physics, general chemistry, and electromagnetism I and II (optional). 3 points.
Magnetic resonance imaging is a fast-growing interdisciplinary field. In this course, students learn how the knowledge they gain from their education in physics, chemistry, mathematics, and computer science can be utilized to further understand the biomedical sciences.

Cryoelectron Microscopy of Macromolecular Assemblies
G16.4408 3 points.
This comprehensive course covers the theory and practice of solving molecular structures by electron microscopy. The course starts with optics, sample preparation, and a basic mathematical description of diffraction before moving into a detailed exploration of the three main methods of structure determination: electron crystallography, single-particle analysis, and electron tomography. The course ends with a discussion of map interpretation and molecular fitting. This is predominantly a lecture course involving one 2-hour lecture per week accompanied by a discussion session and an occasional practical session using the facilities at the New York Structural Biology Center. Lectures are given by expert electron microscopists from around New York City, and students from various campuses are encouraged to attend.

Advanced Magnetic Resonance Imaging
G16.4409 Prerequisite: G16.4404. 6 points.
This course continues from G16.4404, taught in the fall, and successful completion of the fall course is a prerequisite. The course introduces and utilizes mathematical concepts such as the Fourier transform, k-space, and the Bloch equations to describe the physical and mathematical principles governing data acquisition and image reconstruction. Topics covered include diffusion, perfusion, functional brain imaging, cardiac MRI, spectroscopic imaging, clinical MRI, rf engineering, contrast agents, and molecular imaging. The course includes weekly lectures, discussion sessions revolving around assigned research articles, and practical labs pertinent to material covered in the lectures.

Biology

Bioinformatics and Genomes
G23.1127 Identical to G22.2520. Prerequisites: calculus I and II, demonstrated interest in computation, and permission of the instructor. 4 points.
The recent explosion in the availability of genome-wide data, such as whole-genome sequences and microarray data, led to a vast increase in bioinformatics research and tool development. Bio-informatics is becoming a cornerstone for modern biology, especially in fields such as genomics. It is thus crucial to understand the basic ideas and to learn fundamental bioinformatics techniques. The emphasis of this course is on developing not only an understanding of existing tools but also the programming and statistics skills that allow students to solve new problems in a creative way.

Genomics
G23.1128 Prerequisites: V23.0021-0022. 4 points.
Introduction to genomic methods for acquiring and analyzing genomic DNA sequence. Topics: genomic approaches to determining gene function, including determining genome-wide expression patterns; the use of genomics for disease-gene discovery and epidemiology; the emerging fields of comparative genomics and proteomics; and applications of genomics to the pharmaceutical and agbiotech sectors. Throughout the course, computational methods for analysis of genomic data are stressed.

Statistics in Biology
G23.2030 Lecture and laboratory. Prerequisites: college algebra and/or calculus. 4 points.
Advanced course on techniques of statistical analysis and experimental design that are useful in research and in the interpretation of biology literature. Principles of statistical inference, the design of experiments, and analysis of data are taught using examples drawn from the literature. The course covers the use of common parametric and nonparametric distributions for the description of data and the testing of hypotheses.

Computer Science

Fundamental Algorithms
G22.1170 Prerequisites: at least one year’s experience with a high-level language such as Pascal, C, C++, or Java; knowledge of assembly language; and familiarity with recursive programming methods and with data structures (arrays, pointers, stacks, queues, linked lists, binary trees). 3 points.
Reviews a number of important algorithms, with emphasis on correctness and efficiency: solving recurrence equations; sorting algorithms; selection; binary search; hashing; binary search trees and balanced-tree strategies; tree traversal; partitioning; graphs; spanning trees; shortest paths; connectivity; depth first search; breadth first search. Dynamic programming, divide and conquer.

Programming Languages
G22.2110 3 points.
Design and use of mainstream programming languages: naming, scoping, type models, control structures, procedural abstractions, modularization. Considers implementation issues and runtime organization. Languages studied include Ada, C, C++, Java, LISP, ML, and Python. Extensive programming exercises in various languages.

Scientific Computing
G22.2112 Prerequisites: multivariate calculus, linear algebra, and basic probability. C/C++ programming very helpful. 3 points.
A practical introduction to scientific computing, covering theory and basic algorithms, together with the use of visualization tools and principles behind reliable, efficient, and accurate software. Students program in C/C++ or MATLAB. Specific topics include IEEE arithmetic, conditioning and error analysis, classical numerical analysis (finite difference and integration formulas, etc.), numerical linear algebra, optimization and nonlinear equations, ordinary differential equations, and basic Monte Carlo.

Machine Learning
G22.2565 Prerequisites: undergraduate course in linear algebra and strong programming skills for implementation of algorithms studied in class. Recommended: knowledge of vector calculus, elementary statistics, and probability theory. 3 points.
This course covers a wide variety of topics in machine learning, pattern recognition, statistical modeling, and neural computation. The course covers the mathematical methods and theoretical aspects but primarily focuses on algorithmic and practical issues.

Foundations of Machine Learning
G22.2566 3 points.
This course introduces the fundamental concepts and methods of machine learning, including the description and analysis of several modern algorithms, their theoretical basis, and the illustration of their applications. Many of the algorithms described have been successfully used in text and speech processing, bioinformatics, and other areas in real-world products and services. The main topics covered are probability and general bounds, PAC model, VC dimension, perceptron, Winnow, support vector machines (SVMs), kernel methods, decision trees, boosting, regression problems and algorithms, ranking problems and algorithms, halving algorithm, weighted majority algorithm, mistake bounds, learning automata, Angluin-type algorithms, reinforcement learning, and Markov decision processes (MDPs).

Special Topics in Computer Science
G22.3033 3 points.
A selected recent topic is described below.

Computational Biology/ Bioinformatics
The term “computational biology” was originally coined by an analogy to the role that computing has played in the physical sciences. At present, its most obvious role is primarily in terms of gathering, warehousing, and analyzing large amounts of statistical data that can be generated by high-throughput experiments (e.g., whole-genome sequence data, microarray-based gene expression data). It is beginning to be expanded, however, to include many other advances in biology, namely, design of new biotechnology (e.g., single-DNA molecule analysis, nanoscale analysis of biological materials, etc.), creation of novel systems biological models (e.g., WNT signaling, caspase cascade models of intrinsic apoptosis, cell-cycle models, circadian and ultradian cycles, etc.), machine learning approaches to generate hypotheses from data (e.g., a map of cancer, SNP-based haplotype structures, copy-number polymorphisms, etc.), synthesizing new biological objects and systems (e.g., synthetic biology, engineered bacteria, etc.), and many others.

The emphasis of this course is to introduce students to this encompassing view. Topics include introduction to algorithmic biology; some biology for computer scientists; computer science fundamentals; mapping and sequence assembly; sequence analysis; interlude: inference, estimation, and probabilistic analysis; modeling transcription and genomic regulation; structural bioinformatics; pathways: metabolic, signaling, and others; evolution; from polymorphisms (SNPs and CNPs) to disease genetics; and biotechnology of the future.

Mathematics

Numerical Methods I
G63.2010 Identical to G22.2420. Prerequisites: undergraduate linear algebra and some experience with programming. 3 points.
Floating-point arithmetic; conditioning and stability; numerical linear algebra, including systems of linear equations, least squares, and eigenvalue problems; LU, Cholesky, QR, and SVD factorizations; conjugate gradient and Lanczos methods; Gauss quadrature. Current software packages. Computer programming assignments form an essential part of the course.

Special Topics in Numerical Analysis
G63.2011 3 points.
A selected recent topic is described below.

The Immersed Boundary Method for Fluid-Structure Interaction:

The immersed boundary method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry. It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide range of problems in biofluid dynamics, such as wave propagation in the inner ear, fish swimming, and insect flight. Nonbiological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. The main idea of the method is that boundaries, obstacles, or immersed elastic structures can be represented in a unified way in terms of the forces that they apply to the fluid.

Topics to be covered include mathematical formulation of the fluid-structure interaction problem in terms of the Dirac delta function; discretization of the structure, fluid, and interaction equations; methods for handling immersed boundaries with nontrivial mass; methods for immersed filaments with bend and twist; nanoscale hydrodynamics with Brownian motion; adaptive mesh refinement; parallelization; visualization of results as computer animations; and applications. Students have the opportunity to work in teams on computing projects which may involve particular applications and/or proposed improvements in immersed boundary methodology.

Numerical Methods II
G63.2020 Identical to G22.2421. Prerequisite: G63.2010. 3 points.
Nonlinear equations (Newton’s method). Ordinary differential equations: initial value problem (Runge-Kutta and multistep methods, convergence and stability); two-point boundary value problem. Elliptic partial differential equations: finite difference and finite element methods, fast solvers, multigrid, iterative methods exploiting special structure. Brief introduction to time-dependent partial differential equations. Current software packages. Computer programming assignments form an essential part of the course.

Methods of Applied Mathematics
G63.2701 Corequisites: undergraduate advanced calculus, ordinary differential equations, and complex variables. 3 points.
Convergent and divergent asymptotic series; asymptotic expansion of integrals: steepest descents, Laplace principle, Watson’s lemma, and methods of stationary phase; regular and singular perturbations of differential equations, the WKB method, boundary-layer theory, matched asymptotic expansions, and multiple-scale analysis; Rayleigh-Schrödinger perturbation theory for linear eigenvalue problems; summation of series, Pade approximation; averaging methods; renormalization groups; weakly nonlinear waves and geometric optics

Special Topics in Mathematical Biology
G63.2851, 2852 3 points.
Selected recent topics are described below:

Cardiac Mechanics and Electrophysiology

This course is about the equations of a heartbeat, which are partial differential equations. The Navier-Stokes equations of a viscous incompressible fluid, suitably modified to include fluid-structure interaction with the muscular heart walls and the flexible heart valve leaflets, describe the mechanical function of the heart. Cardiac mechanics is coordinated and controlled by an electrical system governed by Hodgkin-Huxley equations in their Bidomain form, which describes both intracellular and extracellular current and voltage, coupled by transmembrane ionic and capacitive currents. Both mechanical and electrical activity are strongly influenced by the fiber architecture of the heart, the differential geometry of which is governed by partial differential equations derived primarily from considerations of mechanical equilibrium. To what extent is the fiber architecture of the heart and its valves determined by these equations? Meanwhile, at the molecular level, the contractile machinery of the heart is described by population dynamics equations that govern the attachment (birth), motion (aging), and detachment (death) of myosin cross bridges interacting with the actin filaments of the muscle. What are the special features of cardiac cross-bridge dynamics that give cardiac muscle its distinctive properties in comparison to skeletal muscle? The emphasis of the course is on the formulation of the detailed realistic models described above and on the numerical solution of the model equations. Simplified models that allow for analytic or asymptotic solution are also introduced for comparison.

Biomolecular Motors

Biological cells contain microscopic robotic machinery that is used for cell motility, for transport of vesicles and organelles within cells, to move protein molecules across internal membranes, to partition chromosomes at cell division, and to manufacture energy-rich compounds such as ATP as well as information-rich compounds such as proteins and nucleic acids. Unlike the macroscopic machinery of everyday experience, these biomolecular motors function in a regime in which Brownian motion (i.e., thermal fluctuation) plays an important role. Throughout the course, mathematical modeling and computer simulation are used to elucidate the diverse mechanisms of biomolecular motors, with particular emphasis on the probabilistic aspect of their function. Topics to be studied include cross-bridge dynamics in muscle, kinesin as a molecular walker, optimal dynamic instability of microtubules for chromosome capture, a depolymerization ratchet mechanism for the movement of chromosomes, the role of elasticity in the function of biomolecular motors, the role of chromosome flexibility in chromosome transport during mitosis, a look-ahead mechanism for RNA polymerase, and rotary molecular motors driven by ion gradients, such as ATP synthase and the bacterial flagellar motor. Students have the opportunity to work in teams on computer simulations of selected motor systems, with computer animation as a means of visualizing the results.

Statistical Analysis of Genomic Data

This course concerns statistical techniques applied to the analysis of large-scale genomic data. Typical data classes considered include genomic sequence data, data from alignments of two or more such sequences, global expression data (e.g., from microarrays), genome annotations, and ontologies. Statistical software (e.g., R) is used, and assignments and projects involve a certain amount of coding and biological interpretation of the results. Students review basic concepts and learn and apply techniques from regression, multivariate analysis, computational statistics, and statistical modeling. Students attend lectures, read recent articles, and work in small, multidisciplinary groups on data analysis assignments and final projects.

Mathematical Neuroscience

This course begins by covering fundamentals of physiological properties of neurons, from neuronal and synaptic dynamics, to rate vs. spike codings. Then it delves into various mathematical aspects of neuronal network modeling, addressing issues of neuronal model reductions (for example, reduction from Hodgkin-Huxley models to integrate-and-fire models), dynamical systems approach, stochastic processes, and nonlinear system analysis in neuronal dynamics. It covers, in detail, a non-equilibrium statistical physics approach to population dynamics of neuronal networks and studies various closures and related kinetic theories. It ends with topics on plasticity and learning. The course strives to bring students with applied mathematics, physical science, or neuroscience background quickly to research topics in theoretical modeling in neuroscience.

Special Topics in Mathematical Physiology
G63.2855, 2856 3 points.
Selected recent topics are described below:

Mathematical Aspects of Neurophysiology

The emphasis of this course is on fundamental mechanisms at the neuron level, i.e., on the building blocks for neural networks. Topics covered include membrane channels (current-voltage relations and gating, including the analysis of patch-clamp data), Hodgkin-Huxley equations (their physical basis, mathematical structure, and numerical solution on the tree-like structure of a neuron), synaptic transmission (including the statistics of vesicle release), and the analysis of neuronal spike trains (including the technique of reverse correlation). Both asymptotic and numerical methods are introduced and explained throughout the course, which can therefore serve as an applied introduction to these methodologies. Students have the opportunity to work individually or in teams on computing projects related to the course material.

Modeling of Neuronal Networks

This course involves the formulation and analysis of models for neuronal ensembles and neuronal computations. Spiking and firing rate mechanistic treatments of network dynamics as well as probabilistic behavioral descriptions are covered. The course considers mechanisms of coupling, synaptic dynamics, rhythmogenesis, synchronization, bistability, and adaptation. Applications likely include central pattern generators and frequency control, binocular rivalry, working memory, decision making and neuro-economics, feature detection in sensory systems, and cortical oscillations (gamma, up-down states). Students undertake computing projects related to the course material.

Chemistry

Mathematical Methods in Chemistry
G25.2626 4 points.
Provides students with the fundamental mathematical tools needed for further study in theoretical chemistry. Topics include vector spaces, linear algebra, ordinary and partial differential equations, special functions, complex analysis, and integral transforms.

Center for Neural Science

Mathematical Tools for Neuroscience
G80.2206 Open to doctoral candidates in fields relevant to neural science. Prerequisites: undergraduate calculus and some programming experience. 4 points.
Team-taught intensive course. Lectures, readings, and laboratory exercises cover basic mathematical techniques for analysis and modeling of neural systems. Homework sets are based on the MATLAB software package.

Simulation and Data Analysis
G80.2233 Identical to G89.2233. Prerequisite: a statistics course, G80.2206, or permission of the instructor. 3 points.
Covers topics in numerical analysis, probability theory, and mathematical statistics essential to developing Monte Carlo models of complex cognitive and neural processes and testing them empirically. Most homework assignments include programming exercises in the MATLAB language.

Linear Systems
G80.2236 Identical to G89.2236. Prerequisite: a semester of calculus or permission of the instructor. 3 points.
Introduction to linear systems theory and the Fourier transform. Intended for those working in biological vision or audition, computer vision, and neuroscience and assumes only a modest mathematical background.