Bioengineering/matlab

Systems Biology Workshop

10/8/2016

A couple of things…

• Voter registration ends tomorrow (10/9) • Group Project

– Pick a partner by Wednesday (10/10) – Will give me partner name on Wednesday – Email me project topic by Friday (10/12) – Project will be due 10/22 (a Monday)

Intro to Systems Biology* • Systems biology: the study of biological function and mechanisms,

underpinning inter- and intra-cellular dynamic networks, by means of signal- and system-oriented approaches

• Systems biology approach means – Investigating components of cellular networks and their interactions – Applying experimental high-throughput techniques – Integrating computational and theoretical methods with experimental efforts

*Dr. Carlo Cosentino – CMU University

• Geneticist: p53 oscillation to regulate the cell cycle

• Chemist/Pharmacology: binding energy of protein-drug complexes

• Mathematician/Engineer: dynamic patterns of pulsatile flow in a heart

What can be modeled?

Biomedical Engineer can technically model all of the above.

Model Behaviors

• Governed by inputs and outputs • Could be qualitative vs. quantitative,

deterministic vs. stochastic, discrete vs. continuous

• Steady state: asymptotic behavior (reversible vs. irreversible)

The modeling process

• Determine the model scope • Select model type • Design and develop model • Model analysis and application

Basic Modeling: Stoichiometric Representation

Substrate 1 Substrate 2

Substrate 3

v1 v2

v3

We can represent this network using linear algebra

This is a stoichiometric network

=

1 0 0

−1 0 −1 1 −1 0 0 0 1

𝑣𝑣1 𝑣𝑣𝑣 𝑣𝑣3 𝑣𝑣𝑣

= 𝑑𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑� 𝑑𝑑𝑑𝑑𝑣

𝑑𝑑𝑑𝑑� 𝑑𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑�

N v 𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑� = 0 At steady state

Linear Algebra Basics Linearly Dependent vs. Independent

1 0 0

0 1 0

1 1 0

1 0 0

0 1 0

0 0 1

x1 x2 x3 x1 x2 x3

x3 = x2 x1 + Dependent Independent

1 0 0 0 1 0 0 0 1

Identity matrix (I)

• A matrix multiplied by its inverse equals I

• IA = A = AI • Must be a square matrix

Reduced Row Echelon Form 1. Leading entries in each row should be 1

• Considered Row-Echelon Form 2. Each leading 1 is the only non-zero in the column

1 −𝑣 1 5 0 1 −1 𝑣 0 0 1 6

1 0 0 19 0 1 0 10 0 0 1 6

Row reduce through a series of basic matrix operations between rows (multiple rows, add rows, interchange rows, etc.)

Definitions

1. Basis: a linearly independent set of vectors x1,…,xn

2. Dimension (dim(S)) : # of vectors forming the basis set of S

3. Rank of matrix : # number of rows that are nonzero in row reduced echelon form

4. Nullity: dim(S) – Rank(S)

rref

Linear Algebra Review Are these vectors linearly independent or linearly dependent?

Row reduce this matrix and find the rank and nullity of this matrix, is it linearly independent or linearly dependent?

0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0

0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 2 2 0 0

1 1 1 -1 1 2 4 3 1 3 9 3 1 4 16 5

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Rank = 4 Nullity = 0 Dimension = 4 Linearly Dependent

Independent Dependent

Metabolic Networks

• Metabolism: biochemical process to acquire energy and materials for cellular growth

• Metabolic flux: the rate of turnover of molecules through a metabolism pathway

• Can describe metabolism by the biochemical reactions in the organism

Example: find write in Nv form in steady state

A B v1 v2 v3 C

D E

v4

v5

v6

v7

• v1 produces A (1) • v2 degrades A (-1) • v4 degrades A (-1) • v3, v5-v7 do nothing to A (0)

V1 V2 V3 V4 V5 V6 V7

A B C D E

1 -1 0 -1 0 0 0

Example: find write in Nv form in steady state

1 -1 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 -1 0 0 0 0 0 0 1 -1 0

V1 V2 V3 V4 V5 V6 V7

= 0

A B v1 v2 v3 C

D E

v4

v5

v6

v7

What does this tell us? 1 -1 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 -1 0 0 0 0 0 0 1 -1 0

V1 V2 V3 V4 V5 V6 V7

= 0

1 0 0 0 0 0 -1 0 1 0 0 0 1 -1 0 0 1 0 0 1 -1 0 0 0 1 0 -1 0 0 0 0 0 1 -1 0

Row Reduced

V1 V2 V3 V4 V5 V6 V7

= 0

What fluxes act on a substrate i.e.

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

= 𝑣𝑣𝑑 − 𝑣𝑣𝑣 − 𝑣𝑣𝑣 = 0

Row reduced echelon form can also tell us more

1 0 0 0 0 0 -1 0 1 0 0 0 1 -1 0 0 1 0 0 1 -1 0 0 0 1 0 -1 0 0 0 0 0 1 -1 0

V1 V2 V3 V4 V5 V6 V7

= 0

• There are 5 rows with leading numbers: Rank = 5 • Total columns = 7, therefore nullity = 7-5 = 2 • You have 2 basis (linearly independent) vectors (nullity) that make up your kernel

(null space) • Essentially: every possible set of steady state flux can be expressed as a linear

combination of these vectors (J) • J = ∑ 𝛼𝛼𝑖𝑖

𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑖𝑖𝑑𝑑𝑛𝑛 𝑖𝑖=1 𝑘𝑘𝑖𝑖

2 columns without a non-leading number: nullity = 2

Work through on board

J = 𝛼𝛼1𝑘𝑘1 + 𝛼𝛼2𝑘𝑘2

Using row reduced echelon form, we can find J

Kinetic Modeling

• System dynamics are described with ODEs

• 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

= f(x1,…,xn ; p1,…,pn; t); – x = substrate/products – p = parameters – t = time

• System state: a snapshot of the system at a given time with sufficient info to predict the state at future times – Set of all possible states = system space

Reaction Kinetics and Thermodynamics

• Purpose of metabolism is the extraction of energy from nutrients

• The breakdown of a pathway (S1  S2) is to show how S1 breaks down into S2

• Law of mass action: Reaction rate of probability of collision – V = v(forward) – v(reverse)

Michaelis-Menten kinetics

Previously Now

S P v

E+S ES E+P

Assumptions 1. E + ES = Constant (E total) 2. [S (t=o)] >> [E] (Briggs and Haldone quasi-steady state) 3. quasi-equilibrium: the reversible conversion of E,S to ES is significantly faster than ES  P+E (k1 and k-1 >> k2)

Reaction Rate v is equal to product formation and negative rate of substrate consumption.

Single substrate, single product reaction

E+S ES E+P k1

k-1 k2

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

= −𝑘𝑘1 𝐸𝐸 𝑑𝑑 + 𝑘𝑘−1[𝐸𝐸𝑑𝑑] 𝑑𝑑𝐸𝐸𝑑𝑑 𝑑𝑑𝑑𝑑

= 𝑘𝑘1 𝐸𝐸 𝑑𝑑 − 𝑘𝑘−1 𝐸𝐸𝑑𝑑 − 𝑘𝑘2[𝐸𝐸𝑑𝑑]

𝑑𝑑𝐸𝐸 𝑑𝑑𝑑𝑑

= −𝑘𝑘1 𝐸𝐸 𝑑𝑑 + 𝑘𝑘−1 𝐸𝐸𝑑𝑑 + 𝑘𝑘2[𝐸𝐸𝑑𝑑]

𝑑𝑑𝐸𝐸𝑑𝑑 𝑑𝑑𝑑𝑑

= 𝑘𝑘1 𝐸𝐸 𝑑𝑑 − (𝑘𝑘−1− 𝑘𝑘2)[𝐸𝐸𝑑𝑑]

𝑑𝑑𝐸𝐸 𝑑𝑑𝑑𝑑

= −𝑘𝑘1 𝐸𝐸 𝑑𝑑 + (𝑘𝑘−1+ 𝑘𝑘2)[𝐸𝐸𝑑𝑑]

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

= 𝑘𝑘2 𝐸𝐸𝑑𝑑

Mass action equations that depict concentration in terms of degradation and production

Upper Glycolysis

Substrates: 1. Glucose 2. Glucose 6-P 3. Fructose 6-P 4. Fructose 1,6-bis P 5. ATP 6. ADP

v1 v2 v3 v4 v5 v6 v7

Glucose Glucose 6-P Fructose 6-P Fructose 1,6-bis P

ATP ADP ATP ADP

v3 v4 v2

v1

v5

v6 v7

1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 -1 -1 0 0 -1 0 -1 0 0 1 0 1 0 1 0 0 -1

v7

What is the stoichiometric matrix?

Su bs

tr at

es : 1. Glucose

2. Glucose 6-P 3. Fructose 6-P 4. Fructose 1,6-bis P 5. ATP 6. ADP

Stoichiometric matrix

1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 -1 -1 0 0 -1 0 -1 0 0 1 0 1 0 1 0 0 -1

v5

v2 v3 v4 v5 v6 v7

v1

%ODE for network y(glu)=v1-v2; y(g6p)=v2-v3; y(f6p)=v3-v4; y(f16p)=v4-v5-v6; y(atp)=-v2-v4+v7; y(adp)=v2+v4-v7;

%Rate Equations v1=k1; %constant v2=k2*y(glu)*y(atp); v3=k3*y(g6p); v4=k4*y(f6p)*y(atp); v5=k5*y(f16p); v6=k6*y(f16p); v7=k7*y(adp);

%k-values k1=1; %glycogen phosphorylase EC:2,4,1,1 k2=0.78; %glucokinase EC:2,7,1,2 k3=0.28; %phosphoglucose isomerase k4=0.21; %6-phosphofructokinase k5=0.0154; %fructose-16-bisphosphate k6= 0.17; %fructose bisphosphate aldolase k7=3.76;

Taken from online database Taken from stoichiometric network

𝑑𝑑𝐸𝐸𝑑𝑑 𝑑𝑑𝑑𝑑

= 𝑘𝑘1 𝐸𝐸 𝑑𝑑 − 𝑘𝑘−1 𝐸𝐸𝑑𝑑 − 𝑘𝑘2[𝐸𝐸𝑑𝑑]

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

= 𝑘𝑘2 𝐸𝐸𝑑𝑑

Glucose Glucose 6-P Fructose 6-P Fructose 1,6-bis P v3 v4 v2

v1

v6

ATP ADP v7

ATP ADP v7

What is needed to solve Michaelis-Menten product formation assuming E+P is irreversible?