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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
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.
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.
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