Special Topics in Computational Biology Lecture #3: Modeling ¦ Bud Mishra Professor of Computer Science and Mathematics (Courant, NYU) Professor (Watson School, CSHL) 2 ¦ 12 ¦ 2002 11/20/2018 ©Bud Mishra, 2002

Goal The goal of this course is to understand, design and create a large-scale computational system centered on the biology of individual cells, population of cells, intra-cellular processes, and realistic simulation and visualization of these processes at multiple spatio-temporal scales. 11/20/2018 ©Bud Mishra, 2002

Why? Such a reasoning system, in the hands of a working biologist, can then be used to gain insight into the underlying biology, design refutable biological experiments, and ultimately, discover intervention schemes to suitably modify the biological processes for therapeutic purposes. The course will focus primarily on two biological processes: genome-evolution and cell-to-cell communication. 11/20/2018 ©Bud Mishra, 2002

Truth in Biology ©Bud Mishra, 2002 Presently, there is no clear way to determine if the current body of biological facts is sufficient to explain a phenomenology. In the biological community, it is not uncommon to assume certain biological problems to have achieved a cognitive finality without rigorous justification. Some ideal candidates for such study would include: prion hypothesis, cell cycle machinery (DNA replication and repair, chromosome segregation, cell-cycle period control, spindle pole duplication, etc.), muscle contractility, processes involved in cancer (cell cycle regulation, angiogenesis, DNA repair, apoptosis, cellular senescence, tissue space modeling enzymes, etc.), signal transduction pathways, circadian rhythms (especially the effect of small molecular concentration on its robustness), and many others. 11/20/2018 ©Bud Mishra, 2002

Computational Approaches Fortunately, in the past, similar issues had been faced by other disciplines: for instance, design of complex microprocessors involving many millions of transistors, building and controlling a configurable robots involving very high degree-of-freedom actuators, implementing hybrid controllers for high-way traffic or air-traffic, or even reasoning about data traffic on a computer network. The approaches developed by control theorists analyzing stability of a system with feedback, physicists studying asymptotic properties of dynamical systems, computer scientists reasoning about a discrete or hybrid (combining discrete events with continuous events) reactive systems ---all have tried to address some aspects of the same problem in a very concrete manner. We believe that biological processes could be studied in a similar manner, once the appropriate tools are made available. 11/20/2018 ©Bud Mishra, 2002

Hybrid Systems ©Bud Mishra, 2002 Biological systems are hybrid—they involve both discrete and continuous dynamics. Example: At the cellular level, cell growth and division in a eukaryotic cell, can be described as a sequence of four processes, each one a continuous process triggered by a set of conditions. At the intercellular level, cell differentiation or apoptosis can be viewed as a hybrid system. 11/20/2018 ©Bud Mishra, 2002

The Cell A cell is a small coalition of a set of genes held together in a set of chromosomes (and even perhaps unrelated extrachromosomal elements). They also have set of machinery made of proteins, enzymes, lipids and organelles taking part in a dynamic process of information processing. In eukaryotic cells the genetic materials are enclosed in the cell nucleus separated from the other organelles in the cytoplasm by a membrane. In prokaryotic cells the genetic materials are distributed homogeneously as it does not have a nucleus. Example of prokaryotic cells are bacteria with a considerably simple genome. 11/20/2018 ©Bud Mishra, 2002

The dynamics of cell: ©Bud Mishra, 2002 The cell cycle ) the set of events that occur within a cell between its birth by mitosis and its division into daughter cells again by mitosis interphase period when DNA is synthesized and mitotic phase The cell division by mitosis (into 2 daughter cells) and meiosis (into 4 gametes from germ-line cells); Working of the machinery within the cell---mainly the ones involving replication of DNA, transcription of DNA into RNA and translation of RNA into protein. 11/20/2018 ©Bud Mishra, 2002

The Cell Cycle: ©Bud Mishra, 2002 In growing cells, the four phases proceed successively, taking from 10-20 hrs. Interphase: comprises the G1, S, and G2 phases. DNA is synthesized in S and other cellular macromolecules are synthesized throughout interphase, roughly doubling cell’s mass. During G2 the cell is prepared for mitotic (M) phase when the genetic material is evenly proportioned and the cell divides. Nondividing cells exit the normal cycle, entering the quiesecent G0 state. M G1 G0 G2 S 11/20/2018 ©Bud Mishra, 2002

Differentiation & Suicide Cellular dynamics controls how a cell changes (or differentiates) to carry out a specialized functions Structural or morphological changes (muscles, neural, skin..) Immune systems: Many cell types come together in organized tissues designed to let the body distinguish self from non-self. Programmed Cell Death/Apoptosis: Condensation of the nucleus. Fragmentation of the DNA. Morphological changes followed by consumption by macrophages. 11/20/2018 ©Bud Mishra, 2002

Modeling Biomolecular Networks Agents and Modes: Species and Processes: There are two kinds of agents: S-agents (representing species such as proteins, cells and DNA): S-agents are described by concentration (i.e., their numbers) and its variation due to accumulation or degradation. S-agent’s description involves differential equations or update equations. P-agents (representing processes such as transcription, translation, protein binding, protein-protein interactions, and cell growth.) Inputs of P-agents are concentrations (or numbers) of species and outputs are rates. 11/20/2018 ©Bud Mishra, 2002

P-agents and S-agents ©Bud Mishra, 2002 S1 Process P1 S2 S3 Process P2 11/20/2018 ©Bud Mishra, 2002

Agents & Modes dx/dt = fqi(x,z), ©Bud Mishra, 2002 Each agent is characterized by a state x 2 Rn and A collection of discrete modes denoted by Q Each mode is characterized by a set of differential equations (qi 2 Q & z 2 Rp is control) dx/dt = fqi(x,z), and a set of invariants that describe the conditions under which the above ODE is valid… these invariants describe algebraic constraints on the continuous state… 11/20/2018 ©Bud Mishra, 2002

Mode Definition ©Bud Mishra, 2002 Modes are defined by the transitions among its submodes. A transition: specifies source and destination modes, the enabling condition, and the associated discrete update of variables. Modes and submodes are organized hierarchically. 11/20/2018 ©Bud Mishra, 2002

Example of a Hybrid System q1 and q2 = two discrete modes x = continuous variable evolving as dx/dt = f1(x) in mode q1 dx/dt = f2(x) in mode q2 Invariants:Associated with locations q1 and q2 are g1(x) ¸ 0 and g2(x) ¸ 0, resp. The hybrid system evolves continuously in disc. mode q1 according to dx/dt = f1(x) as long as g1(x) ¸ 0 holds. If ever x enters the “guard set” G12(x) ¸ 0, then mode transition from q1 to q2 occurs. dx/dt =f1(x) g1(x) ¸ 0 q1 dx/dt =f2(x) g2(x) ¸ 0 q2 G12(x) ¸ 0 G21(x) ¸ 0 11/20/2018 ©Bud Mishra, 2002

dX/dt = synthesis – decay § transformation § transport Generic Equation Generic formula for any molecular species (mRNA, protein, protein complex, or small molecule): dX/dt = synthesis – decay § transformation § transport Synthesis: replication for DNA, transcription of mRNA, translation for protein Decay: A first order degradation process Transformation: cleavage reaction ligand binding reaction Transport: Diffusion through a membrane.. 11/20/2018 ©Bud Mishra, 2002

Model of transcription {X,k, n} F = concentration of an mRNA X = concentration of a TF nXm = Cooperativity coefficient kXm = Concentration of X at which transcription of m is “half-maximally” activated. F(X, kXm, nXm) = Xn/[kn + Xn] Y(X, kXm, nXm) = kn/[kn + Xn]=1 - F(X, kXm, nXm) A graph of function F = Sigmoid Function 11/20/2018 ©Bud Mishra, 2002

Transcription Activation Function 11/20/2018 ©Bud Mishra, 2002

Quorum Sensing in V. fischeri Cell-density dependent gene expression in prokaryotes Quorum = A minimum population unit A single cell of V. fischeri can sense when a quorum of bacteria is achieved—leading to bioluminescence… Vibrio fiscehri is a marine bacterium found both as a free-living organism, and a symbiont of some marine fish and squid. As a free-living organism, it exists in low density is non-luminescent.. As a symbiont, it lives in high density and is luminescent.. The transcription of the lux genes in this organism controls this luminescence. 11/20/2018 ©Bud Mishra, 2002

lux gene + - ©Bud Mishra, 2002 luxR luxICDABEG CRP LuxR Ai LuxA LuxI LuxB + - 11/20/2018 ©Bud Mishra, 2002

Quorum Sensing ©Bud Mishra, 2002 The lux region is organized in two transcriptional units: OL: containing luxR gene (encodes protein LuxR = a transcriptional regulator) OR: containing 7 genes luxICDABEG. Transcription of luxI produces the protein LuxI, required for endogenous production of the autoinducer Ai (a small membrane permeable signal molecule (acyl-homoserine lactone). The genes luxA & luxB code for the luciferase subunits The genes luxC, luxD & luxE code for proteins of the fatty acid reductase, needed for aldehyde substrate for luciferase. The gene luxG encodes a flavin reductase. Along with LuxR and LuxI, cAMP receptor protein (CRP) controls luminescence. 11/20/2018 ©Bud Mishra, 2002

Biochemical Network ©Bud Mishra, 2002 The autoimmune inducer Ai binds to protein LuxR to form a complex C0, which binds to the lux box. The lux box region (between the transcriptional units) contains a binding site for CRP. The transcription from the luxR promoter is activated by the binding of CRP. The transcription from the luxICDABEG is activated by the binding of C0 complex to the lux box. Growth in the levels of C0 and cAMP/CRP inhibit luxR and luxICDABEG transcription, respectively. 11/20/2018 ©Bud Mishra, 2002

Biochemical Network ©Bud Mishra, 2002 Ai C0 luxR LuxA, LuxB LuxI LuxR luxICDABEG CRP LuxC, LuxD, LuxE luxR 11/20/2018 ©Bud Mishra, 2002

Notation ©Bud Mishra, 2002 x0 = scaled population x1 = mRNA transcribed from OL x2 = mRNA transcribed from OR x3 = protein LuxR x4 = protein LuxI x5 = protein LuxA/B x6 = protein LuxC/D/E x7 = autoinducer Ai x8 = complex C0 11/20/2018 ©Bud Mishra, 2002

Evolution Equations… ©Bud Mishra, 2002 dx0/dt = kG x0 dx1/dt = Tc[Y(x8, kC0, nC0) F(cCRP, kCRP, nCRP)+b] – x1/HRNA –kG x1 dx2/dt = Tc[F(x8, kC0, nC0) Y(cCRP, kCRP, nCRP)+b] – x2/HRNA –kG x2 dx3/dt = Tl x1 –x3/Hsp-rAiRx7 x3 –rC0x8 –kG x3 dx4/dt = Tl x2 –x4/Hsp-kG x4 dx5/dt = Tl x2 –x5/Hsp-kG x5 dx6/dt = Tl x2 –x6/Hsp-kG x6 dx7/dt = x0(rAll x4 –rAiRx7 x3+rC0x8) –x7/HAi dx8/dt = rAiR x7 x3 –x8/Hsp –rC0x8-kGx8 11/20/2018 ©Bud Mishra, 2002

Parameters ©Bud Mishra, 2002 Tc nCRP Tl kCRP HRNA nC0 Hsp kC0 Hup b Max. transcription rate nCRP Cooperativity coef for CRP Tl Max. translation rate kCRP Half-max conc for CRP HRNA RNA half-life nC0 Cooperativity coef for C0 Hsp Stable protein half-life kC0 Half-max conc for C0 Hup Unstable protein half-life b Basal transcription rate HAi Ai half-life vb Volume of a bacterium rAll Rate constant: LuxI ! Ai V Volume of solution rAiR Rate constant: Ai binds to LuxI kg Growth rate rC0 Rate constant: C0 dissociates x0max Maximum Population 11/20/2018 ©Bud Mishra, 2002

Remaining Questions ©Bud Mishra, 2002 Simulation: Stability Analysis Nonlinearity Hybrid Model (Piece-wise linear) Stability Analysis Reachability Analysis Robustness 11/20/2018 ©Bud Mishra, 2002