Outline A Biological Perspective The Cell The Cell Cycle Modeling Mathematicians I have known

Cancer Heart disease Neurodegenerative illnesses Molecular Basis of Disease

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.

DNA Information is contained in the primary structure (the sequence of bases). Protein Information is contained at multiple structural levels (primary, secondary, tertiary, quaternary) Forms of Biological Information

Two processes must alternate during eukaryotic cell division Genome must be replicated in S phase Genome must be halved during M phase The Cell Cycle

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 DNA replication Chiasmata Dynamic instabilty Cdc mutantsCell-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 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 Dependent Kinases (CDKs) CDK cyclin substrate ATP P product+ ADP

Cyclin D-CDK4 Cyclin E-CDK2 Cyclin A-CDK2 Cyclin B-CDC2

CDK activity

asyn hours cyclin A E Cdk2 Cyclin and CDK expression as cells re-enter the cell cycle G0 G1 S cell cycle phases

Cyclin D-CDK4 Cyclin E-CDK2 Cyclin A-CDK2 Cyclin B-CDC2 CDK inhibitors

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 ~ cells

Overall properties not predictable from what is known about constituent parts Complexity

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

Notion of mathematical modeling adding value to standard approaches Help to formalize and predict behavior, suggest experiments Bioessays 24, Can Mathematical Modeling Help?

Start from a grocery list of parts Break down large scale systems into smaller functional modules Simulate steady states, oscillations, sharp transitions Modeling the Cell Cycle

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.

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. Questions

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. Models, models, everywhere

vs

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. The Hysteresis Model of Novak and Tyson

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 Hysteresis

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. Experimental System Need pic of xenopus

Steady state cdc2 kinase activity as a function of [cyclin] Black dots=experimental Gray dots=proposed Ti=inactivation threshold 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 Cyclin A-CDK2 Cyclin B-CDC2 p27kip1

CDK cyclin substrate ATP P product+ ADP p27kip1 CDK cyclin Inhibited p27kip1 Model of p27kip1 Function

Cyclin E-CDK2 can phosphorylate p27kip1

Increasing [ATP] Drives p27 Phosphorylation P27-P Time (min)

Inhibited p27kip1 CDK cyclin Switch Active CDK cyclin p27kip1 P P Cell cycle progression Switching between Inhibitor and substrate functions

Mathematical analysis of binary activation of a cell cycle kinase which down-regulates its own inhibitor C.D. Thron

P27 binds and inhibits cyclin E-CDK2 Cyclin E-CDK2 phosphorylates and deactivates p27 This creates a positive feedback loop Experimental Observations

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.

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) You say tomato, I say tomahto