Model of antibiotic action on bacterial growth · Plan: Model of Antibiotic Action on Bacterial...

Model of antibiotic action on bacterialgrowth

Martin R. Evans

SUPA, School of Physics and Astronomy, University of Edinburgh, U.K.

November 15, 2016

Collaborators:Philip Greulich (Edinburgh then Cambridge now Southampton),Matt Scott (Waterloo, Ontario), Rosalind Allen (Edinburgh)

M. R. Evans Model of antibiotic action on bacterial growth

Plan: Model of Antibiotic Action on BacterialGrowth

PlanI Antibiotics: the challenges

II Simple model of dynamics in bacterial cellIII Comparison with experimentsIV Conclusions and thoughts

Reference:P. Greulich, M. Scott, M. R. Evans, R. J. Allen (Molecular Systems Biology2015)


Antibiotics in the 20th Century

Alexander Fleming (1928) :“Fleming was a Scottish bacteriologist and Nobel Prize winner(Medicine, 1945), best known for his discovery of penicillin”

By 1940s penicillin was being mass-produced by the Americandrugs industry.

Leading causes of US deaths:

1900: pneumonia, tuberculosis, diarrhoea1997: heart disease, cancer, strokeMajor contributors to global health in the past century

Major challenges challenges: rise of antibiotic resistance, declinein new drug discoveriesHow a bacterial population acquires resistance is a problemsuitable for physics approachesTo combat resistance need good models for how antibiotics work







Major challenges challenges: rise of antibiotic resistance, declinein new drug discoveriesHow a bacterial population acquires resistance is a problemsuitable for physics approaches

To combat resistance need good models for how antibiotics work







Major challenges challenges: rise of antibiotic resistance, declinein new drug discoveriesHow a bacterial population acquires resistance is a problemsuitable for physics approachesTo combat resistance need good models for how antibiotics work


How do antibiotics work?

Target processes that differ between bacteria and human cells

Inhibition of Cell Wall Synthesisbeta-lactams, vancomycinInhibition of Protein Synthesisaminoglycosides, tetracyclines, chloramphenicol, macrolidesAlteration of Cell MembranespolymyxinsInhibition of Nucleic Acid Synthesisquinolones, metronidazole, rifampicinInterference with metabolismsulfonamides, trimethoprim

Rough Classification: Bacteriostatic and Bactericidal


Bacteria for physicists I

Unicellular prokaryotic organism (no nucleus or organelles)

Active unit - spherocylinder often with flagella acting as propellor

Lots of them!“There are approximately ten times as many bacterial cells in thehuman flora as there are human cells in the body, with the largestnumber of the human flora being in the gut flora, and a largenumber on the skin”.- wikipedia

Form dense aggregates such as biofilmsFrom statistical physics perspective ideal candidate for ‘activematter’ microconstituent: driven, noisy, profuse


Bacteria for physicists II

Exponential growth of bacteria: N(t) = N02t/td = N0eλ0t

λ0 is the growth rate; td is the doubling time

Growth-inhibition curve is λ(a)

IC50 is antibiotic dose required to half the growth rate

Standard wisdom has it that slower growing bacteria (e.g.biofilms) are more resilient to antibiotics.


Our Experiments (Matt Scott)

Grow bacteria under different nutrient conditionsE. coli MG1655, MOPS media with 6 different nutrients and differentgrowth rates.

Glucose (glu) 0.64 ; Glucose + amino acids 1.09; Glucose + aminoacids + nucleotides 1.68Glycerol (gly) 0.40; Glycerol + amino acids 0.85; Glycerol + aminoacids + nucleotides 1.35

Measure the efficacy (IC50) of 4 different antibiotics as afunction of growth rate

Chloramphenicol: bacteriostaticTetracycline: bacteriostaticStreptomycin: bactericidalKanamycin: bactercidal


Growth inhibition for ‘bacteriostatic’ antibiotics


Growth inhibition for ‘bactericidal’ antibiotics


Summary of Experimental Results

Bacteriostatic antibiotics (chlor, tet)

Fast-growing bacteria are more susceptibleGrowth inhibition curve shows gradual decrease

Bactericidal antibiotics (strep, kan)

Fast-growing bacteria are less susceptibleGrowth inhibition curve shows sharp, threshold-like decrease

Why do different antibiotics behave differently?


A simple model

Antibiotic crosses membrane; net inflow rate J.Antibiotic binds ribosomes at rate kon, unbinds at rate koff

Cell grows at rate λ, diluting cell contents (division not explicitlyincluded)New ribosomes (unbound) are synthesized at rate s


Mathematical model IThree variables a, ru, rb describe the state of the cell.Dynamics:

dadt

= −λ(?)a + f (ru, rb,a) + J(aex ,a)

dru

dt= −λ(?)ru + f (ru, rb,a) + s(?)

drb

dt= −λ(?)rb − f (ru, rb,a)

where unbinding/binding rate

f (ru, rb,a) = koffrb − kona(ru − rmin)

and influxJ(aex ,a) = Pinaex − Pouta

How do we model synthesis rate s and growth rate λ?


Empirical Growth Laws

H. Bremer and P. Dennis, in E. coli and Salmonella (1996)

M. Scott, C. W. Gunderson, E. M. Mateescu, Z. Zhang and T. HwaScience (2010) 330, 1099:

1 measured ribosome concentration as function of growth rate (inabsence of antibiotic) and found linear relation

ru = λ0/κt + rmin

i.e. growth rate increases linearly with unbound ribosomeconcentration

2 measured total ribosome concentration as function of growthrate, in presence of antibiotic and found another linear relation

ru + rb = rmax − λC

i.e. The cell upregulates ribosome synthesis to compensate forantibiotic challenge


Mathematical model II: steady state

0 =dadt

= −λa + f (ru, rb,a) + J(aex ,a)(1)

0 =dru

dt= −λru + f (ru, rb,a) + s(2)

0 =drb

dt= −λrb − f (ru, rb,a)(3)

In steady state (2) + (3) s = λ(rb + ru) = λ(rmax − λC)⇒ s(λ)

using second growth law

Then solve (1–3) plus first growth law

ru = λ/κt + rmin

for the four variables λ, a, ru, rb in terms of parameters of cell and aex


Mathematical model III: predictions

(λ

λ0

)3

−(λ

λ0

)2

+

(λ

λ0

)[aex

2 IC∗50

(λ∗0λ0

)+

14

(λ∗0λ0

)2]− 1

4

(λ∗0λ0

)2

= 0

where λ∗0 = 2√

Poutκtkoff/kon is the ‘reversibility parameter’

and IC∗50 =

λ∗0(rmax − rmin)

2Pinis another parameter combination

Setλ

λ0=

12

and solve for aex :

IC50

IC∗50

=12

[(λ0

λ∗0

)+

(λ∗0λ0

)]


Predictions for inhibition curves

Forλ∗0λ0

< 0.3 get hysteresis curve→ discontinuity

Limitλ∗0λ0� 1 λ

λ0=

12

[1−

√1− aex

IC50

]For

λ∗0λ0

> 0.3 get gradual decrease

Limitλ∗0λ0� 1

λ

λ0=

[1 +

aex

IC50

]−1

langmuir-like


Comparison with experiment I

Collapse of data onto limiting growth-inhibition curves - no freeparameters

λ

λ0=

12

[1−

√1− aex

IC50

]λ

λ0=

[1 +

aex

IC50

]−1

Tetracycline and ChloramphenicolStreptomycin and Kanamycin

A B

0 2Relative drug concentration a

ex/IC

50

0

0.2

0.4

0.6

0.8

1

Rela

tive g

row

th r

ate

/0

0 1 2

Relative drug concentration aex

/IC50

0

0.2

0.4

0.6

0.8

1

Rela

tive g

row

th r

ate

/0

Gly Glu GlyCAA

GluCAA

GlyRDM

GluRDM

0.40 0.64 0.85 1.09 1.350(/h) 1.68


Comparison with experiment II

Fit three glycerol curves with two parameters λ∗0 and IC∗50

Chloramphenicol

Kanamycin

A B

DC

Tetracycline

Streptomycin

0 10 20Concentration (µM)

0

0.2

0.4

0.6

0.8

1

Re

lativ

e g

row

th r

ate

λ/λ

0

0 0.1 0.2 0.3 0.4Concentration (µg/ml)

0

0.2

0.4

0.6

0.8

1

Re

lative

gro

wth

ra

teλ

/λ0

0 0.2 0.4Concentration (µg/ml)

0

0.2

0.4

0.6

0.8

1

Re

lative

gro

wth

ra

teλ

/λ0

0 0.5 1 1.5 2Concentration (µM)

0

0.2

0.4

0.6

0.8

1

Re

lativ

e g

row

th r

ate

λ/λ

0


Comparison with experiment III

Universal Curve IC50

IC∗50

=12

[(λ0

λ∗0

)+

(λ∗0λ0

)]

0.1 1 10

λ0/λ

0*

1

10

IC50/C

*

Streptomycin

Kanamycin

Tetracycline

Chloramphenicol


Conclusions

Complex relation between susceptibility and growth environmentcaptured by simple model in which growth and ribosomesynthesis rates depend on ribosome concentration

‘reversibility rate’ λ∗0 provides natural spectrum classification ofribosome-targetting antibiotic actionChloramphenicol and tetracycline (bacteriostatic) ->Fast-growing cells are more susceptible

Streptomycin and kanamycin (bactericidal) ->Fast-growing cells are less susceptible

Further tests of theory by using mutants to vary κt and hence thereduced parameters


Model of antibiotic action on bacterial growth · Plan: Model of Antibiotic Action on Bacterial...

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