Download Fighting Disease with Maths

Fighting Disease with Maths
Sara Jabbari
School of Mathematics & Institute of Microbiology and
Infection, University of Birmingham
9th January 2014
INSTITUTE OF
MICROBIOLOGY
AND INFECTION
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Outline
1
Antibiotics and Resistance
2
Modelling resistance
3
Anti-Virulence Drugs
4
A case study: MRSA
5
Anti-virulence drugs: a general model
6
Summary & future work
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Antibiotics and Disease
Antibiotics are widely used to treat bacterial infections
They act by killing the bacteria, or inhibiting their growth
First discovered in the early 20th Century
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Emergence of resistance
The emergence of antibiotic resistance is becoming an
increasing problem in modern society.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Emergence of resistance
The emergence of antibiotic resistance is becoming an
increasing problem in modern society.
From: Clatworthy et. al, Nat. Chem. Biol., 3(9), 541-548 2007.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Emergence of resistance
The emergence of antibiotic resistance is becoming an
increasing problem in modern society.
From: Clatworthy et. al, Nat. Chem. Biol., 3(9), 541-548 2007.
In the USA, resistant bacteria cause an estimated
∼2 million illnesses p.a.
23,000 deaths p.a.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Acquisition of resistance
There are two principal methods for the acquisition of resistance in
bacteria:
Vertical evolution
Horizontal evolution
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Acquisition of resistance
There are two principal methods for the acquisition of resistance in
bacteria:
Vertical evolution
Spontaneous chromosomal mutation
Horizontal evolution
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Acquisition of resistance
There are two principal methods for the acquisition of resistance in
bacteria:
Vertical evolution
Spontaneous chromosomal mutation
Horizontal evolution
Acquiring genetic material from other resistant organisms
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Acquisition of resistance
There are two principal methods for the acquisition of resistance in
bacteria:
Vertical evolution
Spontaneous chromosomal mutation
Horizontal evolution
Acquiring genetic material from other resistant organisms
Horizontal evolution is thought to be more common, can result in
multi-drug resistance and spreads rapidly, hence it is thought to be
the main concern for antibiotic resistance in a hospital setting.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Modelling the emergence of antibiotic
resistance in-host
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Modelling the emergence of resistance
A - antibiotic
t
eff
an
tib
io
tic
tic
eff
ec
Four variables:
io
tib
an
A
ec
t
conjugation, λ
S
R
m
on
se
im
reversion, ρ
re
sp
e
un
re
sp
un
e
ηR
S - number of susceptible
bacteria
m
se
on
R - number of resistant bacteria
im
ηS
P - number of phagocytes
(immune response)
P
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of a mathematical model
In its simplest form, a mathematical model looks something like:
dN
= births − deaths,
dt
where N is the number of some biological species.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of a mathematical model
In its simplest form, a mathematical model looks something like:
dN
= births − deaths,
dt
where N is the number of some biological species.
In terms of an antibiotic-resistance model:
dR
= birth + conjugation − death − loss of resistance,
dt
where R is the number of antibiotic-resistant bacteria.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of each term: Mass Action Kinetics
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of each term: Mass Action Kinetics
The rate of a reaction is proportional to the concentrations of the
reacting substances.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of each term: Mass Action Kinetics
The rate of a reaction is proportional to the concentrations of the
reacting substances.
e.g . A + B → C
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of each term: Mass Action Kinetics
The rate of a reaction is proportional to the concentrations of the
reacting substances.
e.g . A + B → C
Ô⇒
dC
= rAB,
dt
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Construction of each term: Mass Action Kinetics
The rate of a reaction is proportional to the concentrations of the
reacting substances.
e.g . A + B → C
Ô⇒
dC
= rAB,
dt
dA
= −rAB,
dt
dB
= −rAB.
dt
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
t
ec
dA
= − αA,
dt
dP
=
dt
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
conjugation, λ
S
R
im
reversion, ρ
re
sp
e
n
u
m
im
se
on
m
u
n
e
sp
re
P
dS
=
dt
dR
=
dt
S Jabbari
Fighting Disease with Maths
on
se
ηS
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
t
ec
dA
= 0,
dt
dP
=
dt
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
conjugation, λ
S
R
im
reversion, ρ
re
sp
e
n
u
m
im
se
on
m
u
n
e
sp
re
P
dS
=
dt
dR
=
dt
S Jabbari
Fighting Disease with Maths
on
se
ηS
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
t
ec
dA
= 0,
dt
dP
P
)
= β (S + R) (1 −
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
conjugation, λ
S
R
im
reversion, ρ
re
sp
e
n
u
m
im
se
on
m
u
n
e
sp
re
P
dS
=
dt
dR
=
dt
S Jabbari
Fighting Disease with Maths
on
se
ηS
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
on
se
re
sp
m
u
n
e
im
se
on
Fighting Disease with Maths
R
reversion, ρ
sp
re
dR
=
dt
S
e
n
u
m
dS
=
dt
S Jabbari
t
ec
conjugation, λ
ηS
im
dA
= 0,
dt
dP
P
) − γ(S + R)P
= β (S + R) (1 −
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
S Jabbari
t
ec
− γRP
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
dR
=
dt
R
reversion, ρ
sp
re
− γSP
S
e
n
u
m
dS
=
dt
conjugation, λ
ηS
im
dA
= 0,
dt
dP
P
) − γ(S + R)P
= β (S + R) (1 −
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
− ψR,
S Jabbari
t
ec
− γRP
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
− ψS,
R
reversion, ρ
sp
re
dR
=
dt
− γSP
S
e
n
u
m
dS
=
dt
conjugation, λ
ηS
im
dA
= 0,
dt
dP
P
) − γ(S + R)P − ψP,
= β (S + R) (1 −
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
− ψR,
S Jabbari
t
ec
− γRP
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
− ψS,
R
reversion, ρ
sp
re
dR
=
dt
− γSP
S
e
n
u
m
dS
S
)
= ηS S (1 −
dt
Smax
conjugation, λ
ηS
im
dA
= 0,
dt
P
dP
= β (S + R) (1 −
) − γ(S + R)P − ψP,
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
ǫR
max A
R − γRP
θ+A
S Jabbari
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
− ψR,
−
R
reversion, ρ
sp
re
dR
=
dt
S
e
n
u
m
dS
S
ǫS A
) − max S − γSP
= ηS S (1 −
dt
Smax
θ+A
− ψS,
t
ec
conjugation, λ
ηS
im
dA
= 0,
dt
P
dP
= β (S + R) (1 −
) − γ(S + R)P − ψP,
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
ǫR
max A
R − γRP + λSR
θ+A
S Jabbari
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
− ψR,
−
R
reversion, ρ
sp
re
dR
=
dt
S
e
n
u
m
dS
ǫS A
S
) − max S − γSP − λSR
= ηS S (1 −
dt
Smax
θ+A
− ψS,
t
ec
conjugation, λ
ηS
im
dA
= 0,
dt
P
dP
= β (S + R) (1 −
) − γ(S + R)P − ψP,
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
S
R
ǫR
max A
R − γRP + λSR
θ+A
S Jabbari
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
− ρR − ψR,
−
sp
re
dR
=
dt
reversion, ρ
e
n
u
m
dS
ǫS A
S
) − max S − γSP − λSR + ρR
= ηS S (1 −
dt
Smax
θ+A
− ψS,
t
ec
conjugation, λ
ηS
im
dA
= 0,
dt
P
dP
= β (S + R) (1 −
) − γ(S + R)P − ψP,
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
S
R
− ρR − ψR,
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
R
ǫR A
dR
= cηS R (1 − ) − max R − γRP + λSR
dt
K
θ+A
sp
re
− ψS,
reversion, ρ
e
n
u
m
S
ǫS A
dS
= ηS S (1 −
) − max S − γSP − λSR + ρR
dt
Smax
θ+A
S Jabbari
t
ec
conjugation, λ
ηS
im
dA
= 0,
dt
P
dP
= β (S + R) (1 −
) − γ(S + R)P − ψP,
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
The Model
ec
t
S
R
− ρR − ψR,
Fighting Disease with Maths
on
se
re
sp
m
u
n
e
im
se
on
dR
R
ǫR A
= cηS R (1 − ) − max R − γRP + λSR
dt
K
θ+A
sp
re
− ψS,
reversion, ρ
e
n
u
m
S
ǫS A
dS
= ηS S (1 −
) − max S − γSP − λSR + ρR
dt
Smax
θ+A
S Jabbari
t
ec
conjugation, λ
ηS
im
dA
= 0,
dt
P
dP
= β (S + R) (1 −
) − γ(S + R)P − ψP,
dt
Pmax
eff
an
ti
b
io
ti
c
eff
c
ti
io
b
ti
an
A
P
ηR
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Model Simulation - constant antibiotic
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Model Simulation - antibiotic dosing
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Anti-virulence drugs
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Alternatives to antibiotics
Conventional antibiotics act on bacterial growth, imposing
selective pressure on the bacteria, leading to the emergence of
resistance.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Alternatives to antibiotics
Conventional antibiotics act on bacterial growth, imposing
selective pressure on the bacteria, leading to the emergence of
resistance.
How do we get round this?
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Targeting virulence
Prevent the bacteria from being able to cause harm in the host
until cleared via natural defences:
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Targeting virulence
Prevent the bacteria from being able to cause harm in the host
until cleared via natural defences:
Flushed out of the system
Cleared by the host’s immune response
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Targeting virulence
Prevent the bacteria from being able to cause harm in the host
until cleared via natural defences:
Flushed out of the system
Cleared by the host’s immune response
In theory this should exert little to no selective pressure on the
organism.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Targeting virulence
Prevent the bacteria from being able to cause harm in the host
until cleared via natural defences:
Flushed out of the system
Cleared by the host’s immune response
In theory this should exert little to no selective pressure on the
organism.
Possible mechanisms to target:
cell adhesion
virulence gene regulation
toxin delivery
toxin function
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Staphylococcus aureus
(MRSA)
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Staphylococcus aureus
Methicillin Resistant Staphylococcus Aureus (antibiotic
resistant)
skin infections, pneumonia, sepsis, toxic shock syndrome...
new therapies needed – one target is the quorum sensing
system
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Quorum sensing
cell-cell signalling mechanism
population density dependent behaviour
used in pathogenesis, biofilm formation sporulation, ...
D
Downregulated, non−virulent
U
U
U
U
U
Upregulated, virulent
U
D
D
U
AIP signal molecule
U
U
U
U
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Quorum sensing in S. aureus
AIP
outside the cell
AgrD
AgrB
AgrC
cell membrane
inside the cell
mRNA
AgrA
agrBDCA
P2 P3
S Jabbari
RNA III
Virulence factors
Fighting Disease with Maths
Infection
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Quorum sensing in S. aureus
AIP
outside the cell
AgrD
AgrB
AgrC
cell membrane
inside the cell
mRNA
AgrA
agrBDCA
P2 P3
S Jabbari
RNA III
Virulence factors
Fighting Disease with Maths
Infection
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Quorum sensing in S. aureus
AIP
outside the cell
AgrD
AgrB
AgrC
cell membrane
inside the cell
mRNA
AgrA
agrBDCA
P2 P3
S Jabbari
RNA III
Virulence factors
Fighting Disease with Maths
Infection
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Quorum sensing in S. aureus
AIP
outside the cell
AgrD
AgrB
AgrC
cell membrane
inside the cell
mRNA
AgrA
agrBDCA
P2 P3
S Jabbari
RNA III
Virulence factors
Fighting Disease with Maths
Infection
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Quorum sensing in S. aureus
AIP
outside the cell
AgrD
AgrB
AgrC
cell membrane
inside the cell
mRNA
AgrA
agrBDCA
P2 P3
S Jabbari
RNA III
Virulence factors
Fighting Disease with Maths
Infection
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Mathematical model
Outside the cell
da
= kT S − β Ra + γ R∗ − λa a
dt
Cell membrane
dS
= αS D − δS S − kT S
dt
dT
= αT B − δT T
dt
Inside
the cell
dR∗
= β Ra − (γ + δR∗ )R∗
dt
dAP
= φ AR∗ − (µ + δAP )AP
dt
dA
= κ M − φ AR∗ + µ AP − δA A
dt
dB
= κ M − (αT + δB )B
dt
dD
= κ M − (αS + δD )D
dt
dR
= αRC − β Ra + γ R∗ − δR R
dt
b
dP
= AP (1 − P) − uP
dt
N
dC
= κ M − (αR + δC )C
dt
dM
= Nm + NvP − δM M
dt
S Jabbari
Fighting Disease with Maths
Summary
Anti-Virulence Drugs
(ii)
0
0
5
0
0
10
10
0
0
10
10
R∗ (τ )
10
5
5
10
5
x 10
5
0
0
10
10
0
0
5
5
S Jabbari
10
0
0
10
10
(viii)
5
10
5
10
(xii)
1
500
0
0
x 10
5
(xi)
1000
5
0
0
10
5
5
500
(x)
AP (τ )
5
5
(ix)
5
1000
T (τ )
S(τ )
500
0
0
0
0
500
(vii)
10
0
0
500
(vi)
1000
D(τ )
5
1000
C(τ )
0.5
(v)
(iv)
1000
B(τ )
500
A general model
(iii)
1
A(τ )
M(τ )
1000
a(τ )
(i)
A case study: MRSA
P(τ )
Modelling resistance
R(τ )
Antibiotics
5
10
0.5
0
0
Fighting Disease with Maths
Summary
Anti-Virulence Drugs
(ii)
0
0
5
0
0
10
(v)
5
5
0
0
10
(ix)
10
R∗ (τ )
10
5
0
0
5
10
5
0
0
10
5
x 10
5
0
0
5
5
10
0
0
10
10
(viii)
5
10
5
10
(xii)
1
500
0
0
x 10
5
(xi)
1000
5
0
0
10
10
500
(x)
5
5
1000
T (τ )
500
0
0
0
0
500
(vii)
10
S(τ )
D(τ )
10
500
(vi)
1000
R(τ )
5
1000
C(τ )
0.5
Summary
(iv)
1000
B(τ )
500
A general model
(iii)
1
A(τ )
M(τ )
1000
a(τ )
(i)
A case study: MRSA
P(τ )
Modelling resistance
AP (τ )
Antibiotics
5
10
0.5
0
0
A population of cells will quickly become up-regulated (i.e. virulent)
S Jabbari
Fighting Disease with Maths
Anti-Virulence Drugs
(ii)
0
0
5
0
0
10
(v)
5
5
0
0
10
(ix)
10
R∗ (τ )
10
5
0
0
5
10
5
0
0
10
5
x 10
5
0
0
5
5
10
0
0
10
10
(viii)
5
10
5
10
(xii)
1
500
0
0
x 10
5
(xi)
1000
5
0
0
10
10
500
(x)
5
5
1000
T (τ )
500
0
0
0
0
500
(vii)
10
S(τ )
D(τ )
10
500
(vi)
1000
R(τ )
5
1000
C(τ )
0.5
Summary
(iv)
1000
B(τ )
500
A general model
(iii)
1
A(τ )
M(τ )
1000
a(τ )
(i)
A case study: MRSA
P(τ )
Modelling resistance
AP (τ )
Antibiotics
5
10
0.5
0
0
A population of cells will quickly become up-regulated (i.e. virulent)
How can we prevent this?
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Synthetic inhibition
AIP
AgrD
AgrB
mRNA
INHIB
AgrC
AgrA
agrBDCA
P2 P3
S Jabbari
RNA III
Virulence factors
Fighting Disease with Maths
Infection
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Synthetic inhibition - numerical solutions
0.9
1
0.9
Proportion of
up−regulated cells
Steady−state proportion of
up−regulated cells
0.8
2
0.8
kINHIB=10
0.7
k
0.6
k
0.5
0.4
3
=10
INHIB
3
=5×10
INHIB
3
kINHIB=6.5×10
0.3
k
0.2
4
=10
INHIB
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0.7
0
10
20
τ
30
40
50
0
0
0.5
k
1
1.5
INHIB
S Jabbari
Fighting Disease with Maths
2
4
x 10
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Synthetic inhibition - numerical solutions
0.9
1
0.9
Proportion of
up−regulated cells
Steady−state proportion of
up−regulated cells
0.8
2
0.8
kINHIB=10
0.7
k
0.6
k
0.5
0.4
3
=10
INHIB
3
=5×10
INHIB
3
kINHIB=6.5×10
0.3
k
0.2
4
=10
INHIB
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0.7
0
10
20
τ
30
40
50
0
0
0.5
k
1
1.5
INHIB
2
For sufficiently large dosage, synthetic inhibitor therapy can
successfully downregulate the cells, but the outcome may be
dependent upon the initial conditions.
S Jabbari
Fighting Disease with Maths
4
x 10
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
How can we improve this?
What are the key properties of a
successful inhibitor?
Steady state proportion of
up−regulated cells
1
0.8
0.6
0.4
0.2
0
0
1
2
3
k
4
5
i
6
7
10
x 10
kcrit =
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
How can we improve this?
What are the key properties of a
successful inhibitor?
Steady state proportion of
up−regulated cells
1
0.8
0.6
0.4
Minimise the rate of separation.
0.2
0
0
1
2
3
k
4
5
i
6
7
10
x 10
(λ + γi )λi γi k̂a v̂ 4 (P̄U† − P̄U† )
3
kcrit =
4
(λ + γ)λa u(k̂S v̂ P̄U† + λ)
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Consistent with experimental results
From: J.S. Wright et al. (2005) PNAS 102: 1691-1696
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
MRSA: a summary
MRSA controls production of its virulence factors in
accordance with its population size (quorum sensing)
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
MRSA: a summary
MRSA controls production of its virulence factors in
accordance with its population size (quorum sensing)
We can block quorum sensing to reduce virulence factor
production
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
MRSA: a summary
MRSA controls production of its virulence factors in
accordance with its population size (quorum sensing)
We can block quorum sensing to reduce virulence factor
production
BUT this may only be successful if the infection is caught
sufficiently early
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
MRSA: a summary
MRSA controls production of its virulence factors in
accordance with its population size (quorum sensing)
We can block quorum sensing to reduce virulence factor
production
BUT this may only be successful if the infection is caught
sufficiently early
Will this help combat antibiotic resistance?
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
A general model of antibiotics and
anti-virulence drugs
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Anti-virulence drugs
conjugation, λ
S
R
im
reversion, ρ
ηR
im
,γ
m
se
on
un
e
sp
re
re
sp
e
un
on
se
m
,γ
ηS
P
antimicrobial effect
antimicrobial effect
A∗
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
A mathematical model for anti-virulence drugs
dA∗
= 0,
dt
dP
P
= β (S + R) (1 −
) − δSR (S + R) P − δP P,
dt
Pmax
⎛
dS
S
γmax A∗ h ⎞
= ηS S (1 − ) − γ +
PS − λSR + ρR
dt
K
⎝
γ50 h + A∗ h ⎠
− ψS,
⎛
R
γmax A∗ h ⎞
dR
= (1 − c) ηS R (1 − ) − γ +
PR + λSR
dt
K
⎝
γ50 h + A∗ h ⎠
− ρR − ψR.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Anti-virulence drugs simulation
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Anti-virulence drugs simulation
Resistant bacteria eliminated.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Anti-virulence drugs simulation
Resistant bacteria eliminated.
Small level of susceptible
bacteria remain at infection site.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Anti-virulence drugs simulation
Resistant bacteria eliminated.
Small level of susceptible
bacteria remain at infection site.
Why?
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Patient-specific parameters
ψ, general clearance rate
γ, baseline immune response
With strong host clearance
mechanisms treatment can be
completely effective.
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Patient-specific parameters
ψ, general clearance rate
γ, baseline immune response
With strong host clearance
mechanisms treatment can be
completely effective.
Not suitable therefore for
already hospitalised patients?
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Combining Antibiotics and Anti-Virulence Drugs
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Combining Antibiotics and Anti-Virulence Drugs
Both drugs administered
simultaneously
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Combining Antibiotics and Anti-Virulence Drugs
Both drugs administered
simultaneously
Susceptible bacteria cleared
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Combining Antibiotics and Anti-Virulence Drugs
Both drugs administered
simultaneously
Susceptible bacteria cleared
Small population of resistant
bacteria remain
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Combining drugs with a time delay
Introduction of antimicrobial first
followed by antibiotic after t = 15
days.
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Combining drugs with a time delay
Introduction of antimicrobial first
followed by antibiotic after t = 15
days.
S Jabbari
Introduction of antibiotic first
followed by antimicrobial after
t = 9 days.
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Combining drugs with a time delay
Introduction of antimicrobial first
followed by antibiotic after t = 15
days.
Introduction of antibiotic first
followed by antimicrobial after
t = 9 days.
Complete bacterial elimination achieved
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Summary
Mathematical modelling and differential equations can be used
to represent the inner-workings of bacteria, infection and the
effects of drugs
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Summary
Mathematical modelling and differential equations can be used
to represent the inner-workings of bacteria, infection and the
effects of drugs
Anti-virulence drugs show promise for the future, but possibly
only either in combination with antibiotics, or for the
prevention of treatment
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Summary
Mathematical modelling and differential equations can be used
to represent the inner-workings of bacteria, infection and the
effects of drugs
Anti-virulence drugs show promise for the future, but possibly
only either in combination with antibiotics, or for the
prevention of treatment
Future work:
Refine the models for specific bacteria and infections, e.g. E.
coli in urinary tract infections
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Summary
Mathematical modelling and differential equations can be used
to represent the inner-workings of bacteria, infection and the
effects of drugs
Anti-virulence drugs show promise for the future, but possibly
only either in combination with antibiotics, or for the
prevention of treatment
Future work:
Refine the models for specific bacteria and infections, e.g. E.
coli in urinary tract infections
Use the models to determine which aspects of the
anti-virulence drugs should be improved and the optimal
treatment strategies
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Summary
Summary
Mathematical modelling and differential equations can be used
to represent the inner-workings of bacteria, infection and the
effects of drugs
Anti-virulence drugs show promise for the future, but possibly
only either in combination with antibiotics, or for the
prevention of treatment
Future work:
Refine the models for specific bacteria and infections, e.g. E.
coli in urinary tract infections
Use the models to determine which aspects of the
anti-virulence drugs should be improved and the optimal
treatment strategies
Develop to account for the possibility of resistance to
anti-virulence drugs
S Jabbari
Fighting Disease with Maths
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
What can maths do for biology?
Investigate systems that are impractical in the lab
e.g. the potential for drugs that haven’t yet been developed
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
What can maths do for biology?
Investigate systems that are impractical in the lab
e.g. the potential for drugs that haven’t yet been developed
Accelerate and reduce experimental work
e.g. identify most promising aspects on which to focus
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
What can maths do for biology?
Investigate systems that are impractical in the lab
e.g. the potential for drugs that haven’t yet been developed
Accelerate and reduce experimental work
e.g. identify most promising aspects on which to focus
Predict optimal strategies
e.g. dosing strategies
S Jabbari
Fighting Disease with Maths
Summary
Antibiotics
Modelling resistance
Anti-Virulence Drugs
A case study: MRSA
A general model
Thank you!
,
Any questions??
S Jabbari
Fighting Disease with Maths
Summary