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CHAPTER II UNDERSTANDING BIOCHEMICAL SYSTEM FOR PATHWAYS RECONSTRUCTION Hiren Karathia (Ph.D- System Biology and Bioinformatics) Supervisor: Dr. Rui Alves Ref from: Prof. Michael A. Savageu and Dr. Claudio cobelli, David Foster and Gianna Toffolo 14 th January 2010

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INDEX SYSTEMS UNDERSTANDING OF COMPARTMENTAL KINETICS NON-COMPARTMENTAL KINETICS KINETICS PARAMETERS OF THE ABOVE KINETICS

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Two types of pools are seen in estimating kinetics parameter in systems biology study 1.Single pool system study 2.Multipool system study

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Systems Accessible Pool concept Accessible pool is part of a large system from where variables of system could be accessed. Such System’s variables could be analyzed for various parameters of system and estimate for its weight in whole system. Two types of pools are seen in estimating kinetics parameter in systems biology study 1.Single pool system study 2.Multipool system study SYSTEM ENVIRONMENT UNIVERSE A B c

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Single Pool System For simplicity we will use single pool as single chemical species or Protein. Single protein analysis is done with respect to measurements (concentration or mass), de novo production transport within system, interaction with other protein or chemical (other pool interaction [multipool analysis]) within a system and disposal from the system. SYSTEM ENVIRONMENT UNIVERSE A De novo production Disposal A

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Multipool system In multipool system single or more than one protein either perform function of transformation, exchange, recreation or interactions.

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Overall view of Non-compartmental system CVCV CVCV K1 K1’

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Kinetic Parameters for Non compartmental kinetics Mass M (units: mass): This is the mass of material in the accessible pool, i.e. the pool in which the samples measuring its amount will be taken. Volume of distribution V (units: volume): This is the volume of the accessible pool. It is a volume in which the Protein (chemical species) is uniformly distributed. Clearance rate CR (units: vol /time): This is the rate at which the accessible pool is irreversibly cleared of material per unit time.

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Kinetic Parameters for Non compartmental kinetics Fractional clearance rate FCR (units: 1/time): This is the fraction of material that is irreversibly lost from the accessible pool per unit time. (The FCR is sometimes called the fractional catabolic rate.) Mean residence time (units: time): This is the average time a particle spends in the accessible pool during all passages through it before leaving it for the last time. Rate of appearance Ra (units: mass/ time): This is the rate at which the material enters the accessible pool for the first time. Rate of disappearance Rd (units: mass/ time): This is the rate at which the material is irreversibly lost from the accessible pool.

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The relationships among the parameters

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Michaelis-Menten Kinetics where k1 and k2 are rates of forward reaction and k-1 is rate of reverse reaction. E = free enzyme S = the free substrate P = free product ES = Enzyme substrate complex At particular time ‘t’ free concentration of enzyme Et is total number of free and bound enzymes. Et = (ES) + E

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Michaelis-Menten Kinetics The rate of product to form is v = dP/dt = k2(ES) It is assumed that rate of product is directly proportional to what ever form of ES form at that time. d(ES)/dt = k1ES – (k-1 + k2) (ES) At steady state ES complex rate becomes constant so 0 = k1ES – (k-1 + k2) (ES) Et = E + (ES)

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Michaelis-Menten Kinetics Solution of this equation is given by

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Michaelis-Menten Kinetics

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Monomolecular Kinetics for mono molecular kinetics Intra molecular interactions or cleavage of a molecule, the only type of reaction that involve a single substrate, are called monomolecular. The above is depends on the thermal energy to make in a given time interval is proportional to exp(-E A /RT) OR In concentration term –dA/dt = kA (unit is reciprocal time)

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Bimolecular reaction X1 + X2 -> X3 The probability of reaction is proportional to two factors: 1.Mono molecular face – required sufficient amount of thermal energy to yield a reaction 2.Collision between X1 and X2 molecules dX3/dt = kX1X2 = -dX1/dt = -dx2/dt

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Trimolecular reactions When more than one molecules are involved, the assumption is made that the reaction can be broken into stages, each having only two reacting molecules. dX4/dt = kX1X2X3 = -dX1/dt = -dX4/dt = -dX4/dt

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Modification of Reactions Enzyme inhibition: Four types of inhibitions are possible in system to inhibit Enzyme, Substrate reactions. 1.Competitive 2.Uncompetitive 3.Mixed 4.Partially competitive

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COMPETITIVE INHIBITION An inhibitor that interact with the same group on enzyme as a normal substrate and blocking subsequent activity. Possible due to similarity of steric configuration between substrate and inhibitor (I). Inhibitor and Substrate compete for a single site to bind on Enzyme. i.e., EI complex cannot make ES and vice versa at the same time on single enzyme molecule.

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COMPETITIVE INHIBITION The rate equation for the mechanism is derived by assuming steady state assumption. V M = k 2 E t, K m = (k -1 +k 2 )/k 1 and k 3 = k -3 /k 3. In reciprocal form

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Graph of Competitive inhibition Slope is a linear function of concentration (I), while interception on 1/v is independent.

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Uncompetitive inhibition Inhibitor do not block substrate specific site on the enzyme. The rate equation Where Vm = k2Et, Km = (k-1 + k2)/k1 and k3 = k-3/k3

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Graph of Non competitive inhibition First graph is family of curves for different concentration of I. Slope is independent of the inhibition concentration.

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Mixed Inhibition Inhibitor which interact with the enzyme at a site distinct from the substrate specific site. Binding of substrate or inhibitor to the enzyme modifies subsequent affinity for inhibitor or substrate respectively.

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Rate equations

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Compartmental System’s model This model is to “compartmentalize” the system, i.e. to postulate a structure for the no accessible portion of the system consisting of distinct “compartments” which are interconnected by pathways representing fluxes of material and/or biochemical conversions.

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COMPARTMENTAL VIEW UNIVERSE ENVIRONMENT EXCHANGE OF ENERGY AND CONCENTRATION SYSTEM [M1] V1 C3 V3 C2 V2 K1 K1’ K2 K2’ K3’ K3

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Other view with Compartmental model

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Explanation The compartmental method relaxes the limits imposed by the no compartmental model. It will provide the investigator with insights into the system structure, permitting predictions about components of the system not accessible for measurement. In fact, it is not possible to track the behaviour of every molecule in a biological system at every point in time. Hence it is useful to consider collections of specific molecules at specific sites or in specific forms, i.e. collections of molecules having similar characteristics but existing in the system at different locations, or collections that exist at a given site or location in the system but have different characteristics.

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Example Once glucose is transported from plasma to muscle cells, it can be phosphorylated to glucose-6-phosphate. Thus muscle cells are a location in the glucose system where glucose molecules are present in two different forms, glucose and glucose-6-phosphate. The system in these terms, i.e. collecting molecules at specific sites or in specific forms, permits a “lumping” of the system into discrete groups, and that arrows can be used to represent the movement from one site or one form to another.

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Basis of Compartmental Modeling The basis of the compartmental system, or model, are the arrows which are used to indicate the interconnections among the various compartments. As will be seen, the interconnections represent a flux of material (mass) which, physiologically, represents transport or a chemical transformation or both.

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Distribution of material in Compartmental model Amount of material that acts as though it is well-mixed and kinetically homogeneous in a Compartment

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Concentration of substance In (A) mass of substance is non uniformly distributed in a system. The concentration of substance depends upon where the sample is taken. Thus Sample S1 and S2 would have different concentration in a compartment. In (B) mass of substance is uniformly distributed. Two sample taken from different sites in the system would have same concentration.

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Well Mixed Well mixed in compartments means any two samples taken from the compartment at the same time would have the same concentration of the substance being studied and therefore be equally representative.

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Kinetic homogeneity Every particle in a compartment has the same probability of taking the pathways leaving the compartment. There are several pathways by which material can leave. Each pathway may and probably will have a different probability; the sum of all of the probabilities is, of course, equal to 1.

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Kinetic homogeneity Every compartment is characterized both by an amount of material and what can happen to that material. Basically, material flows into and out from the compartment, and the balance between the two determines the amount of material in the compartment at any point in time. Thus when material leaves a compartment, it does so because of metabolic event related to transport and utilization. For a given compartment, there may be several such events possible. It is the totality of such events that characterize the behaviour of material in a compartment. Kinetic homogeneity means that each particle of the material in the compartment have the same probability of leaving due to one of these events.

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Discrete nature of compartment A compartment, therefore, is a discrete amount of material that behaves identically. The discrete nature of a compartment is what allows one to reduce a complex biological system into a finite number of discrete compartments and pathways.

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Kinetic Parameters of Compartmental Kinetics

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For any substance in a compartment dM/dt = -F 01 + U 1 = 0 Kij is defined as Rate constant or “Fractional Transfer Coefficient” For one compartment model mass balance equation is

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Two compartment Model

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Mass Balance for compartment For two compartment, dM1/dt = -F01 – F21 + F12 + U1 = - (K01 + K21)M1 + K12M2 + U = 0 dM2/dt = -F 01 – F 21 + F 12 + U1 = - (K 01 + K 21 )M1 + K 12 M2 + U2 = 0

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N compartmental Model the description of a general the n- compartment system is more complex. The mass balance equations, however, are obvious extensions of those given for the two compartment system. Given in terms of the rate constants K ij they are

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N compartmental Model

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Matrix form

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