Pathway Analysis Exemplified with Models of Dopamine Metabolism Eberhard O Eberhard O. Voit it Department of Biomedical Engineering Georgia Institute of Technology and Emory University 2012 Winter School in Mathematical and Computational Biology St. Lucia, Queensland, 2-6 July 2012
Overview 1. Metabolic Pathways are Good Modeling Targets Advantages Steps of Generic Model Design Diagnostic Methods Methods of Analysis 2. Example: Dopamine Metabolism Importance of the Pathway Specific Steps of Model Design Model Structure Results Implications for Diseases and Treatment 2
Metabolic Pathways as Modeling Targets Simplified Central Dogma: Genes mRNA Protein Metabolites Response (Disease) Signaling Structure Other Functions (cell cycle, translation, ….) A whole lot can happen between gene expression and an organismic response! Also, metabolic pathways obey stoichiometric rules
Example: Dopamine Signaling Neurotransmitter Signals sent from one neuron (presynapse) to another (postsynapse) across a synapse. Human Brain: between 100 and 500 trillion synapses Presynapse Synapse Postsynapse psicopolis. com
Neurotransmitter Dopamine • Signals sent from midbrain to forebrain • Involved in motor control, reward, learning, addiction (amphetamine) 5
Dopamine-Associated Diseases Parkinson’ s Disease Systemic disease Symptoms: resting tremor, rigidity, postural instability, loss of smell, … Diagnosis: experience based, subjective, no biomarker Etiology: environmental, genetic factors, and their interactions Pathogenesis: oxidative stress, mitochondrial dysfunction, protein misfolding, dopamine loss Pathology: early onset vs. later onset Treatment: mainly symptom relief, side effects, loss of effectiveness 6
Dopamine-Associated Diseases Schizophrenia Mental Disease Symptoms: abnormalities in perception or expression of reality Diagnosis: self-reported and/or psychiatric Etiology: environmental, genetic factors, recreational and prescription drugs, …, interactions Pathogenesis: Could be one or several disorders, increased dopamine activity Attention Deficit/Hyperactivity Disorder, Autism Drug Addiction 7
Dopamine-Associated Diseases Macroscopic Level: Motor dysfunction, depression, mood, shaking, loss of smell, … Neuron loss Allostasis Mesoscopic Level: Function and malfunction in the presynapse Function and malfunction in the postsynapse Function and malfunction in circuitry Microscopic Level: Genetic predisposition Electrophysiology and properties of membranes Enzyme kinetics Mechanisms of drug action Details of brain circuitry 8
Major Modeling Challenge: Pathway System Ill-defined Typical situation in human disease: Not all metabolites and enzymes are known Not all regulatory signals are known Parameters are uncertain Concentrations are uncertain Flux rates are uncertain Q: Is there enough information to start constructing a model? 9
Scope, Goals, Objective Scope, Goals, Objective Generic Goals, Inputs, Data, Information and Initial Modeling Prior Knowledge Exploration Strategy Model Type of Model Selection Variables, Interactions Model Design Equations, Parameter Values Consistency, Robustness Model Analysis Validation of Dynamics and Diagnosis Exploration of Possible Behaviors Hypothesis Testing, Simulation, Model Discovery, Explanation Use and Applications Manipulation and Optimization
Scope, Goals; Data 1. Understand dopamine dynamics in the presynapse 2. Understand function of dopamine in the postsynapse 3. Understand normal signal transduction across a synapse 4. Characterize pathological conditions; including effects of drugs 5. Suggest means of therapeutic intervention 6. Lots known about DA-associated diseases (macro-scale) 7. Some knowledge about DA metabolism; but quantitative details rare 8. Some knowledge about signal reception and interpretation 11
Modeling Context Meth- Amphetamine Presynapse DA synthesis Vesicle dynamics DA recycling Postsynapse Incoming signals Changes DARPP-32 AMPAR
Type of Model Needs to be dynamic Could be discrete-time Choose continuous time; differential equations Although interesting spatial components, ignore space; at least initially Choose ordinary differential equations (ODEs) Alternative could be agent-based models (ABMs) Although stochastic effects, ignore them initially for simplicity ODEs much simpler than corresponding stochastic models 13
Variables and Processes (Pre /synapse*) Biochemistry: Metabolite concentrations; enzymatic reactions 14 * Postsynapse later
Translate Diagram into Equations Generic Strategy: + – V i V i dX i X V V X i i i i dt V V ( X , X ,..., X , X ,..., X ) i i 1 2 n n 1 n m inside outside Solution with Potential: n f / “Biochemical Systems Theory” V X k , i , j ik i , k j (BST) j 1 15
BST Formulation Example: 3,4-dihydroxyphenylacetaldehyde (DOPAL) in Synapse d DOPAL / dt = V + (Dopamine, MAO, SSAO, H 2 O 2 ) – V - (DOPAL, ALDH) = Dopamine g 1 MAO g 2 SSAO g 3 H 2 O 2 g 4 – DOPAL h 1 ALDH h 2
Alternative Formulations Within BST S-system Form : g h g g h h i , n m i , n m X X X ... X X X ... X i 1 i 2 i 1 i 2 i i 1 2 n m i 1 2 n m + – V i 1 V i 1 dX i X V V X i i ij ij dt – + V i ,q V i ,p 17
Alternative Formulations Within BST S-system Form : g h g g h h i , n m i , n m X X X ... X X X ... X i 1 i 2 i 1 i 2 i i 1 2 n m i 1 2 n m + – V i 1 V i 1 dX i X V V X i i ij ij dt – + V i ,q V i ,p Generalized Mass Action Form : f X X ijk i ik j 18
Notable Mathematical Features of BST Steady-state equations of S-systems linear Recasting: Equivalence transformations of any ODE system into S-system format; function classification Interesting limit cycle / Hopf bifurcation analysis De novo creation of limit cycle oscillators Lie group analysis: Decoupling of systems Statistics: Recast S-system representation of distributions; Approximate S-distribution Facilitated optimization 19
S-system Steady-State Equations Linear g h g g h h i , n m i , n m X X X ... X X X ... X 0 i 1 i 2 i 1 i 2 i i 1 2 n m i 1 2 n m Define Y i = log( X i ): log g Y g Y g Y i i 1 1 i 2 2 i , n m n m log h Y h Y h Y i i 1 1 i 2 2 i , n m n m 1 1 Y A b A A Y D D D I I S-system highly nonlinear, but steady-state equations linear. 20
Where are We in the Process? Scope, Goals, Objective Scope, Goals, Objective Goals, Inputs, Data, Information and Initial Prior Knowledge Exploration Variables and processes Model Type of Model Selection directly from diagram; equations from BST Variables, Interactions Model Need to determine Design Equations, Parameter Values parameter values: , , g , h , initial values Consistency, Robustness Model Analysis Validation of Dynamics and Diagnosis Exploration of Possible Behaviors Hypothesis Testing, Simulation, Model Discovery, Explanation Use and 21 Applications Manipulation and Optimization
Challenge: Parameter Estimation Parameter: A quantity in a function or set of equations that remains constant during a mathematical evaluation (“computational experiment”), but may vary from one experiment to the next. Parameter Estimation (Mathematics): The process of identifying values of parameters in a model that (typically) minimize the difference between the output of the model and corresponding data. Example: F ( x ) = m x + b Issue: Have determined suitable functional form. How do I find the best parameter values?
Parameter Estimation Strategies Bottom up: Find kinetic parameters for each enzyme (process) Convert information into numerical values of all parameters Merge all representations, hope for the best; refine Often, this method requires MANY iterations Top down Use time course data of many or all variables Apply an optimization algorithm to fit the model to data Data of this type are rare “Other” 23
Traditional Bottom-up Strategy dX j V i = R i ( S i , M i ) = f k ( X j , V i ) dt p 1 , p 2 , p 3 , … 24 Voit , Drug Discovery Today , 2004
Estimation Based on Time Series and BST BST g h X X X g ' h ' Y ' Y ' Y g '' h '' Z ' ' Z ' ' Z 25 Voit , Drug Discovery Today , 2004
Here: “Other” Typical situation, especially in human (disease) systems: Not all metabolites and enzymes are known Not all regulatory signals are known Kinetic and regulatory characteristics are uncertain Concentrations are uncertain Flux rates are uncertain 26
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