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Outline •A Biological Perspective •The Cell •The Cell Cycle •Modeling •Mathematicians I have known Molecular Basis of Disease Cancer Heart disease Neurodegenerative illnesses If we can understand disruption of molecular events at the cellular level we can perhaps prevent or stop disease manifestation at the organismal level The Cell A variety of membrane-bounded compartments exist within eucaryotic cells, each specialized to perform a different function. Forms of Biological Information DNA •Information is contained in the primary structure (the sequence of bases). Protein •Information is contained at multiple structural levels (primary, secondary, tertiary, quaternary) The Cell Cycle Two processes must alternate during eukaryotic cell division •Genome must be replicated in S phase •Genome must be halved during M phase Cell cycle events must be highly regulated in a temporal manner Genetic and molecular studies in diverse biological systems have resulted in identification and characterization of the cell cycle machinery Mitotic spindle Chiasmata DNA replication Dynamic instabilty Cell-cycle control Maturation-promoting factor Regulation of Cdc2 Cyclin characterization Checkpoint control p53 The mitotic checkpoint The APC and proteolysis SCF and F-box proteins Cdc mutants The restriction point Yeast centromeres Cell-cycle conservation Replication origins Retinoblastoma/E2F Body-plan regulation A new class of cyclins CDK inhibitors Sister-chromatid cohesion The cell cycle engines cyclin CDK •Cyclin Dependent Kinases (CDKs) P substrate product ATP + ADP Cyclin D-CDK4 Cyclin E-CDK2 Cyclin A-CDK2 Cyclin B-CDC2 CDK activity Cyclin and CDK expression as cells re-enter the cell cycle asyn 0 4 8 12 14 16 20 24 28 32 hours cyclin A cyclin E Cdk2 G0 G1 S cell cycle phases G2/M S G1 cyclin E protein Cdk2 bound to cyclin E cyclin E associated kinase act iv ity 1 2 3 4 5 6 7 8 Cyclin D-CDK4 Cyclin E-CDK2 CDK inhibitors Cyclin B-CDC2 Cyclin A-CDK2 The Cell Cycle •Complex system •Components are identified •Highly regulated •Defined parameters Cell Cycle Characteristics •Temporally ordered events •Irreversibility •Oscillations •Checkpoints •Positive and negative feedback loops Positive Feedback Loop 70kg human ~ 1013 cells Complexity Overall properties not predictable from what is known about constituent parts Reductionist-analytical strategies focus on component properties and actions, but do not necessarily describe dynamic behavior of the larger system. The best test of our understanding of cells will be to make quantitative predictions about their behavior and test them. This will require detailed simulations of the biochemical processes taking place within cells… Hartwell, Hopfield, Leibler, and Murray What’s the problem? •Cartoons are cartoons •They do not quantitatively describe the experimental data they summarize •Used in a loose qualitative manner •Informal, verbal •Not reliable for judging accuracy of mechanistic proposals Can Mathematical Modeling Help? •Notion of mathematical modeling adding value to standard approaches •Help to formalize and predict behavior, suggest experiments Bioessays 24, 2002. Modeling the Cell Cycle •Start from a grocery list of parts •Break down large scale systems into smaller functional modules •Simulate steady states, oscillations, sharp transitions •Formulate interactions as precise molecular mechanisms. •Convert the mechanism into a set of nonlinear ordinary differential equations. •Study the solutions of the differential equations by numerical simulation. •Use bifurcation theory to uncover the dynamical principles of control systems. Cells progressing through the cell cycle must commit irreversibly to mitosis. Questions •What causes cyclin degradation to turn on and off periodically? •Why don’t rates of synthesis and degradation balance each other? •There must be some mechanism for switching irreversibly between phases of net cyclin synthesis and net cyclin degradation. Models, models, everywhere •Many competing models because the degrees of freedom were unbounded. •Could occur by hysteresis (ie toggle-like switching behavior in a dynamical system). •Time delayed negative feedback loops. vs The Hysteresis Model of Novak and Tyson •Describes a network of interlocking positive and negative feedback loops controlling cell cycle progression. •Proposes a bistable switch is created by the positive feedback loops involving cyclin B-cdc2 and its regulatory proteins. Hysteresis •It takes more of something to push a system from state A to B than it does to keep the system in B. •Creates a bistable system with a rachet to prevent slippage backwards. •Irreversibility was proposed to arise on transversing a hysteresis loop Experimental System Need pic of xenopus •Using Xenopus egg extracts to demonstrate the cell cycle exhibits hysteresis •The amount of cyclin required to induce entry into mitosis is larger than the amount of cyclin needed to keep the extract in mitosis. Steady state cdc2 kinase activity as a function of [cyclin] Black dots=experimental Ti=inactivation threshold Gray dots=proposed Ta=activation threshold The hysteresis model made nonintuitive predications that were confirmed experimentally. • [cyclin B] to drive mitosis > [cyclin B] to stay in mitosis. • Unreplicated DNA elevates the cyclin B threshold for cdc2 activation; ie checkpoints enlarge the hysteresis loop. • Cdc2 activation slows down at cyclin B concentrations marginally above the threshold. Mathematicians I have known Cyclin D-CDK4 Cyclin E-CDK2 p27kip1 Cyclin B-CDC2 Cyclin A-CDK2 cyclin CDK Model of p27kip1 Function P substrate product ATP cyclin CDK p27kip1 p27kip1 Inhibited + ADP Cyclin E-CDK2 can phosphorylate p27kip1 wtp2 7 p2 7 (8 7 -1 9 8) - + cyclin E wt 8 7 -1 98 - + No pre-incubation p27 pr e-incubati on ti me ti me p27 No pre-incubati on phos. p27 p27 pre-incubation 0 5 10 time (min) 15 20 Two Distinct Binding Modes between p2 7 and Cyclin E-Cdk2 loose binding tight binding rapid slow E-K2 -p2 7 E-K2 + p2 7 rapid (E-K2 -p2 7 )* (K2E-p27)tight Inhibitory Interaction Slow (K2E-p27)loose p27 ATP Fast (K2E-p27-ATP)loose K2E ATP (K2E-ATP) K2E + p27P p27 K2E-p27P + ADP Catalytic Cycle Increasing [ATP] Competes with Inhibition of Cyclin E-Cdk2 by p27 Increasing [ATP] Drives p27 Phosphorylation 5000 1000 M ATP 4000 P27-P 500 M ATP 3000 phos. HH1 2000 150 M ATP 1000 50 M ATP 0 0 5 10 time (min) Time (min) 15 20 Switching between Inhibitor and substrate functions P P Switch cyclin CDK cyclin CDK p27kip1 p27kip1 p27kip1 Inhibited Active Cell cycle progression Mathematical analysis of binary activation of a cell cycle kinase which down-regulates its own inhibitor C.D. Thron Experimental Observations •P27 binds and inhibits cyclin E-CDK2 •Cyclin E-CDK2 phosphorylates and deactivates p27 •This creates a positive feedback loop Is the release of EK2 binary (all-or-none)? •Binary enzyme activation implies an abrupt switch from a stable steady state with a low level of free active enzyme. •Implies a bistable system. Small parameter change causes low activity steady state to be extinguished in a saddle-node bifurcation. •Mathematical analysis of the biochemical kinetics required for binary activation. Conclusions An enzyme that attacks and deactivates its own inhibitor is not released from inhibitor binding in an all-or-none fashion unless certain kinetic features are present. You say tomato, I say tomahto •If you want to communicate with someone, you need to speak their language •Convert math to cartoons •Seek out collaborations/sabbaticals •The burden of proof is on you •Look outward as well as inward (kinetics and physiology)