Structural Properties of Networks Systems and Synthetic Biology 424 (2011) Copyright © 2011: Sauro 1 Network Equation Rates of Change Network Structure (Stoichiometry Matrix) Reaction Rate Vector 2 Stoichiometry Matrix 3 Stoichiometry Matrix Relationships among the columns -Flux Analysis Relationships among the rows -Moiety Analysis 4 Conserved Moieties In the above cycle the molar amount of the round subgroup remains constant. Subgroups whose molar amounts remain constant during the evolution of a pathway are called moieties. 5 Conserved Moieties Also called: Conserved Cycles Or Moiety Conserved Cycles 6 Conserved Moieties AMP ADP ATP 7 Conserved Moieties AMP ADP ATP 8 Conserved Moieties Fast AMP Fast ADP Slow ATP Over long timescales the moieties are not conserved. Synthetic and Degradation 9 Conserved Moieties Protein Phosphorylation ATP ADP Phosphate Protein 10 Conserved Moieties Protein Phosphorylation ATP ADP Fast + = T Slow Synthetic and Degradation 11 Examples 1. ATP, ADP, AMP 2. NADH, NAD 3. CoA, Acetyl-CoA 4. Protein, Phosphorylated Protein 5. etc. 12 Examples 1. ATP, ADP, AMP 2. NADH, NAD 3. CoA, Acetyl-CoA 4. Protein, Phosphorylated Protein So what? 13 Examples 1. ATP, ADP, AMP 2. NADH, NAD 3. CoA, Acetyl-CoA 4. Protein, Phosphorylated Protein 1. Cycles add redundancies to the model equations. 2. They introduce a new parameter into the model: Total moles in the cycle 3. Can cause numerical instability in certain calculations (Jacobian becomes non-invertible) 4. Can have profound effects on the behavior of models 14 Redundancies 15 Redundancies 16 Redundancies The sum of S1 and S2 remains constant in time 17 Redundancies The sum of S1 and S2 remains constant in time 18 Redundancies The number of equations has been reduced by one Blackboard 19 New Parameters in the Model The number of equations has been reduced by one 20 New Parameters in the Model Et = E + ES Vmax In a simple cycle, the total molarity in the cycle acts as a linear scaling factor. New Parameters in the Model K A B P Three Conservation Cycles: A/B/C K P C New Parameters in the Model K Increase A B P Three Conservation Cycles: A/B/C K P C New Parameters in the Model K Increase A B P Three Conservation Cycles: A/B/C K P C New Parameters in the Model Sequestration of K K Increase A B C P Three Conservation Cycles: A/B/C K P The net effect is that increases in A inhibit the B -> C reaction. Similar arguments apply to C New Parameters in the Model A B C The end result is a toggle switch that can display two steady states How many are there? For large metabolic maps, 5% to 10% of the species are involved in moiety conserved cycles. That is 5% to 10% of the differential equations are redundant. http://www.iubmb-nicholson.org/ 27 How are conserved cycles identified? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. 28 Linearly Dependent Rows A linearly dependent set of vectors is one where at least one of the vectors can be written as a linear combination of the other vectors. This means there is redundant information in the vector set. 29 How are conserved cycles identified? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. v1 v2 S1 S2 30 The Rank The Rank of a set of vectors arranged in a matrix is the number of linearly independent vectors in the matrix. The Rank for this matrix is two 31 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. What’s the Rank of this matrix? Type rank(N) in Matlab or Jarnac. 32 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. Rank(N) = 2 33 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. Rank(N) = 2 E * -1 = ES S2 = -1 * (ES + S1) 34 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. The Rank for this matrix is two 35 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. The Rank for this matrix is two 36 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. 37 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. 38 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. 39 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. 40 How many conserved cycles are there? Conserved cycles in a pathway appear as linearly dependent rows in the stoichiometry matrix. 41 Echelon Forms There is a particular kind of matrix that one frequently encounteres in the study of the stoichiometry matrix: The Reduced Echelon Form Examples of Matrices in Reduced Echelon Form 1. 2. 3. 4. All rows that consist entirely of zeros are at the bottom of the matrix. In each non-zero row, the first non-zero entry is a one, called the leading one. The leading 1 in each row is to the immediate right of all leading 1’s above it. Each column that contains a leading one has zeros above and below it. If not, then the matrix is in echelon form, otherwise it is in reduced echelon42form Echelon Forms There is a particular kind of matrix that one frequently encountered in studying the stoichiometry matrix: The Reduced Echelon Form Examples of Matrices in Reduced Echelon Form A reduced echelon form has the following general block structure: 43 Echelon Forms Any matrix can be transformed to its reduced echelon form by a series of: 1. Interchanging rows (exchange) 2. Multiplying a row throughout by a nonzero number (scaling) 3. Adding one row, one or more times to another row (replacement) Reduction to reduced echelon form can be used to find conservation laws. 44 Echelon Forms Let us start with this system: Recast in Matrix Form: Identity Matrix 45 Echelon Forms Let us start with this system: Reduce the stoichiometry matrix to reduced echelon form. To preserve the right-hand side, we apply the same operations to the left-side. 46 Echelon Forms Let us start with this system: Partition the M matrix in to X and Y, along the same row line as the reduced echelon form: 47 Echelon Forms Multiplying out the lower partition one obtains: This is the answer we seek. It describes relationships among the rates of change that do not change in time (the definition of a conserved cycle). 48 Let’s Try an Example 49 Let’s Try an Example 50 Let’s Try an Example 51 Let’s Try an Example 52 Let’s Try an Example Y Partition 53 Let’s Try an Example Independent Species Dependent Species Y Partition 54 Let’s Try an Example Independent Species Dependent Species Y Partition I 55 Let’s Try an Example Independent Species Dependent Species Expressing the dependent species in terms of the independent species. 56 Advanced Analysis Independent Species Dependent Species Reorder the rows in the stoichiometry matrix so that the top m_o rows include the independent species and the bottom m – m_o rows include the dependent species. Since the bottom No dependent rows can be derived by linear combinations of the top Nr rows, we can define a matrix, the link matrix (Lo), that can carry out this operation: m = number of rows in the stoichiometry matrix. 57 Advanced Analysis 58 Advanced Analysis Si are the independent species, and Sd the dependent species. 59 Advanced Analysis 60 Advanced Analysis “Y” Matrix = cf. Slide 554 61 Advanced Analysis “Y” Matrix = cf. Slide 55 62 Advanced Analysis 63 Advanced Analysis 64 Advanced Analysis The conservation laws can be obtained by computing the null space (kernel) of the transpose of the stoichiometry matrix. 65 What Happens in Simulators 1. From the stoichiometry matrix obtain: m and the set of dependent and independent species 2. From obtain In your ODE solver: Use compute: and to compute dSi/dt 66 Advanced Analysis 67 Practical Consequences 68 Sleeping Sickness Parasitic Protozoa Trypanosoma, causative agent for an often fatal disease called sleeping sickness. Half a million people are thought to have the disease at any one time. Sleeping Sickness – Chagas’ disease Reduviid bug, otherwise known as the assassin bugs” or “kissing bugs” Treatment The availability of drugs for the treatment of sleeping sickness is poor: 1. Eflornithine is effective for the late stage of the disease and is very expensive. 2. Pentamidine and Suramin have serious side effects. 3. Melarsoprol is very toxic, if not fatal in the wrong dose. Finding Suitable Targets In general, there are two ways to kill or impair a parasite metabolically: 1. Reduce an important flux to a very low value 2. Cause some intermediate metabolite/protein to accumulate to such an extent that is becomes toxic Physiology when in Bloodstream Tryps depends entirely on glycolysis for all energy needs, it has neither a functional Krebs cycle or oxidative phosphorylation. It doesn’t even have a capacity to store carbohydrate. While in the blood, it consumes glucose equivalent to it’s own body mass every hour. Glycolytic rates are probably of the order of 10 times host rates. Conservation Laws There are four conservation laws: NAD AMP (Gly) AMP (Cyt) Phosphate http://bip.cnrs-mrs.fr/bip10/eise2.htm http://bip.cnrs-mrs.fr/bip10/eise3.htm#2 Nature (1996) Supplement on intelligent drug design, 384 (suppl.), 1-26. Singular Jacobian Matrix Blackboard, simple cycle example 75 Newton Raphson Method or How p.ss.eval works Refer to other notes 76