Discovery and Assesment of New Target Sites for
Anti-HIV Therapies
Problem given by:
Sanjive Qazi, Gustavus Adolphus College, U.S.A.
Working group:
Chris Breward, Math. Inst., University of Oxford, U.K.
Jane Heffernan, York University, Canada.
Robert M. Miura, New Jersey Institute of Technology, U.S.A.
Neal Madras, York University, Canada.
John Ockendon, OCIAM Math. Inst. , University of Oxford, U.K.
Mads Peter Sørensen, DTU Mathematics, Tech. Univ. of Denmark.
Bob Anderssen, CSIRO, Mathematical and Information Sciences, Australia.
Roderick Melnik, Wilfrid Laurier University, Canada.
Mark McGuinness, Victoria University, New Zealand.
Fields-MITACS Industrial Problem-Solving Workshop
August 11 – 15, 2008
Introduction
The HIV viruses infect cells by endocytosis and takes over parts of the
cells reaction pathways in order to reproduce itself and spread the
infection.
One such pathway is the mammalian inflammatory signaling, which
invoke NF-κB as the principal transcription factor.
A treatment against HIV could be based on blocking the NF-κB
pathway by a suitably designed drug.
The aim of the current project is to investigate the feasibility of this
idea by using mathematical modelling of the NF-κB pathway.
Fields-MITACS Industrial Problem-Solving Workshop
August 11 – 15, 2008
Outline
 Cartoon model of the inflammatory pathway.
 How HIV attacks mammalian cells through e.g. TNF signalling.
 The role of IKK and the TNF receptor in the cell membrane.
 Mathematical model of the NF-κB pathway.
 The role of IKK signaling. Fixed points and stability.
 Numerical examples.
 Extended mathematical model.
 Fixed points and stability.
 Numerical examples.
 Outlook and further work.
Fields-MITACS Industrial Problem-Solving Workshop
August 11 – 15, 2008
HIV Life Cycle
Protease
HIV
Fusion
Reverse Transcriptase
Viral RNA
Viral RNA
Transcribed to
DNA
RNA + Viral
Proteins
Released
Viral DNA
Incorporated
Into Host
Genome
New Proteins
from Viral DNA
Budding of
New Virion
CD4
receptor
Protease
Enables
Capsid
Assembly
CD4 T-cell
4
Drug Therapy
HIV
Protease
Fusion
Reverse Transcriptase
Viral RNA
Viral RNA
Transcribed to
DNA
RNA + Viral
Proteins
Released
Viral DNA
Incorporated
Into Host
Genome
New Proteins
from Viral DNA
Budding of
New Virion
CD4r
Protease
Enables
Capsid
Assembly
CD4 T-cell
5
Cartoon of the NF-κB pathway
Fields-MITACS Industrial Problem-Solving Workshop
August 11 – 15, 2008
Reaction scheme
k
1


Ac 
 An
k
-1
where
A c =NF B in the cytoplasm
k
2  B +2A +DNA
2A n +DNA 
n
n
A n =NF B in the cell nucleus
Bc =I B (inhibitory  B)in the cytoplasm
k
3 B
Bn 
c
Bn =I B in the cell nucleus
AB=NF B+I B complex
k
4  AB
Bc +A c 
D=I K=inhibitory  B kinase
k1 , k -1 , k 2 , k 3 , k 4 , k 5 are kinetic rate constants
k
5  A +D
AB+D 
c
Mathematical description
Use law of mass action for each of the reactions
Assume constant concentration of D, and combine with k5
We get An after the fact from A=A*-Ac-AB
Mathematical description
Parameter values come from literature (means that someone else
guessed them!)
k4  0.5M 1s 1
A*  0.1M
4
k1  0.8 10 s
1
k1  0.9 101 s 1
k2  1.6 102 s 1
k2  0s 1
k3  2 104 s 1
k5  1.13 104 s 1
Steady State
2

k
k
A
k1 An
3 2 n
A

AB

 c
k3  k2 k5
k1

A*  Ac  An  AB
k3k2 k1 An
k2 An2
Bc 
Bn 
k4 k1 k3  k2 
k3  k  2
Has unique physical fixed point for all positive
parameter values.
Stable at given parameter values
(in general: Jacobian at fixed point has positive
determinant, negative trace, no positive real
eigenvalues).
10



Numerics
2k5
k5=0
11
Numerics
2k5
k5=0
12
Cartoon number 2 of the NF-κB pathway
Modified reaction scheme
where
k
1


Ac 
 An
k
-1
k
2  B +2A +DNA
2A n +DNA 
n
n
k
3 B
Bn 
c
k
4  AB
Bc +A c 
k
5  A +D
AB+D a 
c
i
k
6


TNF+D 0 
 D a +TNF
k
-6
A c =NF B in the cytoplasm
A n =NF B in the cell nucleus
Bc =I B (inhibitory  B)in the cytoplasm
Bn =I B in the cell nucleus
AB=NF B+I B complex
D a =active I K=(inhibitory  B kinase)
D i =inactive I K
D 0 =neutral I K
TNF=tumor necrosis factor
k1 , k -1 , k 2 , k 3 , k 4 , k 5 , k 6 , k -6 are kinetic rate
constants
Modified reaction scheme cont’d
D0 is produced at the constant rate kp and degrades at the rate Li
D1 degrades
TNF is produced outside of the cell in response to HIV infection
Mathematical description
Use law of mass action for each of the reactions
Concentration of TNF is rolled up into k6
Mathematical description
Steady states and stability
• Has unique physical fixed point for the given
parameter values, as well as for all smaller
(nonnegative) values of k6) and  k5).
• Stable at given parameter values (other values
not checked).
Numerics
19
Numerics
20
Numerics
21
Future Work
27 variable model
Systematic reduction to see if it corresponds with our 7 variable model
Control model
Consider problem as optimal control with mu and lambda as the control
parameters
Unclear what to minimize
Sensitivity analysis
Vary rates
22
Reference: Cheong et.al. Understanding NF-κB signaling via mathematical modeling, Molecular Sytems
Biology 4:192, 2008.
Reference: Krishna et.al. Minimal model of spiky oscillations in NF-κB signaling, PNAS 103(29), 1084010845, 2006.
Reference: Chan et.al. Quantitative ianalysis of human immunodeficiency virus type 1-infected CD4+ cell
proteome: … Journal of Virology, 7571-7583, 2007.
Reference: Lipniacki et.al. Mathematical model of NF-κB regulatory module, Journal of Theoretcal Biology
228, 195-215, 2004.
Fields-MITACS Industrial Problem-Solving Workshop
August 11 – 15, 2008
Summary and outlook
HIV viruses take over host cellular pathways for their reproduction. One
such pathway is the NF-κB pathway.
Cartoon modeling of the NF-κB pathway.
Mathematical modeling for clearifying the underlying regulatory pathway
dynamics and hopefully summarizing abundant experimental observations.
Mathematical modling as a tool for rational guided drug targeting.
Extended complex models and mode reduction of bio chemical complexity.
Fields-MITACS Industrial Problem-Solving Workshop
August 11 – 15, 2008