Transcript Document

Introduction to Modelling Signalling
Cascades in Yeast
Jörg Schaber
www.sysbiolab.net
Institute for Experimental Internal Medicine
Otto-von-Guericke University
Magdeburg
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Outline
• The modelling process
• A simple step-by-step example
– The Sho1 branch of the HOG pathway
• Basic concepts
– Signalling motifs
– Model selection
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Why Models?
• Modelling requires verbal hypothesis be made
specific and conceptually rigorous.
• Modelling highlights gaps in our knowledge.
• Modelling provides quantitative as well as
qualitative predictions.
• Modelling is ideal for analysing complex
interactions before experimental tests.
• Modelling is a low-cost, rapid test bed for
candidate interventions.
• Well designed models are readily portable and
adaptable for many purposes.
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The Modelling Process in 8 Steps
Observation
Prior knowledge
Prediction
Hypothesis
Analysis
Word Model
Validation
Math Model
Verification
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A Step-by-Step Example
The Sho1-branch of the HOG pathway
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The Data
• It was shown that
a) Sho1 de-oligomerizes upon osmotic shock
b) Hog1 phosphorylates de-oligomerized Sho1
c) Phosphorylated Sho1 is less able to transmit the
signal
sho1D ssk1D
Oligomer-deficient
Oligomer-deficient
Phospho-mimic
Hao et al. (2007) Curr. Biol. 17
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The Hypothesis
– Phosphorylation of Sho1 by Hog1
constitutes a negative feedback loop.
– This negative feedback leads to the rapid
attenuation of Hog1 signalling.
– Might be important to dampen crosstalk to
pheromone signalling pathway.
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The Word Model(s)
• Signal: high osmolarity
• Hog1 de-sensitizes Sho1
Hao et al. (2007) Curr. Biol. 17
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Formalising Word Models
‘Biologist’ notation
‘Systems Biologist’ notation
Pbs2P
Pbs2P
v1
Hog1
Not very useful (for
modelling), because
interactions not clear.
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Hog1
v2
Hog1P
More useful, because each
interaction is made specific.
This facilitates mathematical
formulation.
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Formalising Word Models
Pbs2P
v1
Hog1
v2
Hog1P
• arrows between components
indicate transformations, i.e.
biochemical reactions, mass
flows. They determine changes
in concentrations, numbers, etc.
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• Instead of giving explicit
formulas for components,
e.g. HogP(t)=f(v1,v2), we
rather characterise the
change of components
over time.
• Change = what goes in
– what goes out
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Formalising Word Models
Pbs2P
v1
Hog1
v2
Hog1P
• That is one reason, why
ordinary differential equation
(ODE) models are so popular:
easy to set up given a carefully
set up wiring scheme:
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dHog1P
 v1  v2
dt
dHog1
 v1  v2
dt
Note that Pbs2P does
not change.
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Formalising Word Models
Pbs2P
v1
Hog1
v2
Hog1P
Biochemical notation
Hog1 + Pbs2P -> Hog1P + Pbs2P
• arrows on arrows indicate
modifying interactions
(enzymatic reactions), i.e. no
(net) mass flows or
concentrations changes
involved from emanating
components.
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Pbs2P neither
consumed nor
produced (netto-wise)
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Formalising Word Models
Most simple
mathematical
formulation: mass
action kinetics, i.e.
multiplication of
substrates.
Pbs2P
v1
Hog1
v2
Hog1P
Biochemical notation
v1: k1·Hog1·Pbs2P
v2: k2·Hog1P
v1: Hog1 + Pbs2P -> Hog1P + Pbs2P
v2: Hog1P -> Hog1
dHog1P
 v1  v2
dt
dHog1
 v1  v2
dt
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Formalising Word Models
• Set up wiring scheme, where each considered reaction
with modifiers is made explicit.
• Formulate ODE system with balance equations.
• Choose kinetics rate formulation.
• Choose initial conditions.
 Theory tells us:
a) There exists a solution.
b) The solution is unique.
c) The solution can be arbitrarily approximated.
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Formalising Word Models
Most simple
mathematical
formulation: mass
action kinetics, i.e.
linear multiplication of
substrates.
Fus3
v1
Ste5
Fus3Ste5
v2
v1: k1·Fus3·Ste5
v2: k2·Fus3-Ste5
Biochemical notation
v1: Fus3 + Ste5 -> Fus3-Ste5
dFus3
 v1  v2
dt
dSte5
 v1  v2
dt
dFus3Ste5
 v1  v2
dt
v2: Fus3-Ste5 -> Fus3 + Ste5
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Kinetic rate laws and signalling motifs
Constant flux
and simple mass action degradation
v1
P
v2
k1
dP
 k 1  k2 P
dt
P (t  0)  0
k2
0  k 1  k2 P
P
k1
k2
Modified constant flux
and simple mass action degradation
S
v1
v3
P
v2
dS
  k3 S
dt
dP
 k 1 S  k2 P
dt
P (t  0)  0
S (t  0)  1
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Kinetic rate laws and signalling motifs
Modified conversion with signal degradation
Mass action kinetics
S
v3
v1
Q
v2
P
dS
  k3 S
dt
dP
 k 1 S (1  P )  k 2 P
dt
Q  P 1
Modified conversion with signal degradation
Michaelis-Menten kinetics
S
v3
v1
Q
v2
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P
k3
k3/2
k3 S
dS

dt
Km3  S
Km3
dP k 1 S (1  P)
k2 P


dt Km1  1  P Km2  P
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Kinetic rate laws and signalling motifs
Modified conversion with signal degradation
Hill kinetics
S
v3
v1
Q
v2
P
k3 S h3
dS

dt
Km3h3  S h3
k 1 S h1 (1  P)
dP
k 2 P h2


h1
h1
dt Km1  (1  P)
Kmh2 2  P h2
k3
k3/2
Km3
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Kinetic rate laws and signalling motifs
Inhibited conversion with signal degradation
S
v3
v1
Q
v2
P
dS
  k3 S
dt
dP
(1  P )
 k1
 k2 P
h
dt
1  Ki S
1/(1+KiSh)
1
h>1
1/2
h=1
ℎ
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1/𝐾𝑖
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Kinetic rate laws and signalling motifs
Modified conversion with negative feedback
S
v1
Q
v2
P
dP
(1  P)
 k1 S
 k2 P
h
dt
1  Ki P
Observation:
• System reaches new
steady state.
• No ‘overshoot’
Possible explanation:
• Feedback comes too
fast.
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Kinetic rate laws and signalling motifs
Modified conversion with delayed negative feedback
S
v1
Q
v2
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P
dP
(1  P)
 k1 S
 k2 P
h
dt
1  K i P(t   )
Observation:
• System ‘overshoots’,
but still reaches new
steady state.
• damped oscillations
• The feedback to P
depends directly from P
itself -> ‘auto-inhibitory
feedback’
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Modified conversion with negative
feedback
Let’s do the math: Calculate steady-states of P(S) of a simplified system
(all constants = 1)
dP S (1  P )

P
dt
1 P
In the steady-state:
(1  P)
0S
P
1 P
1
 P  - 1 - S  1  6S  S 2
2

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
Our analysis suggests
that in a system with
auto-inhibitory negative
feedback, the steadystate depends on the
input signal.
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The Mathematical Model
• Signal: high osmolarity
• Hog1 de-sensitizes Sho1
• Delayed auto-inhibitory
feedback
• Michaelis-Menten kinetics (20
parameters)
• Model was fitted to Hog1
activation data
Hao et al. (2007) Curr. Biol. 17
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Validation
Single shock
Double shock
Orig. 1 M KCl
Orig. 0.5 M KCl
Orig. 0.25 M KCl
- Model fits data well.
- Simulations show damped oscillations and increasing
steady states
- possible spurious effects due to over-parameterization
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Predictions
0.4 M KCl tripple shock (0, 30, 60 min)
Orig. P-Hog1
Orig. Sho1i
Orig. Sho1a
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Conclusion after first modelling round
• Auto-inhibitory feedback model fits single and double
shock data well, but
• shows increasing steady states with increasing
external osmolarity (not supported by data),
• shows damped oscillations (not supported by data, but
might be due to over-parameterization)
• Auto-inhibitory feedback model is not able to predict triple
shock experiment, because of desensitization.
Possible solution: a) the signal has to be removed by adaptation
b) no desensitization
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Recalling the biology
• Increasing the ambient
osmotic pressure leads to a
rapid passive loss of water
and cell skrinkage
• This leads to closure of the
glycerol channel and
activation of two parallel
signaling branches that both
activate Hog1.
• Activated Hog1 translocates
to the nucleous and triggers
production of enzyme that
enhance glycerol prodction.
Pbs2 P
Hog1 P
Gpd1
Glycerol
Fps1
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• Increased glycerol equilibrates
water potential differences and
forces water back into the cell
leading to volume adaptation and
HOG pathway deactivation.
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New Hypothesis
– The signal is the water potential difference
(differences in osmolarity) rather than
merely external osmolarity.
– The main feedback is via glycerol
accumulation.
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The new word model
• Signal = OuterOmolarity-Glycerol
• 3 reactions with mass actions, i.e.
3 parameters
OuterOsmolarity
Glycerol
Signal
v3
v1
Hog1
P-Hog1
v2
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The mathematical model
OuterOsmolarity
Glycerol
Signal
v3
v1
Hog1
P-Hog1
– For simplicity we assume
Hog1+Hog1P=1
– OuterOsmolarity (O) fixed
input function.
v2
dHog1P
 k1 (O  G )(1  Hog1P )  k 2 Hog1P
dt
– The feedback depends on
dG
 k3 Hog1P
increasing Glycerol, which
dt
stays even if Hog1P=0
𝐻𝑜𝑔1𝑃0 , 𝐺0 = (0, 𝑂0 )
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Verification of the model
Let’s do the math: Calculate steady-states
0  k1 (O  G)(1  Hog1P)  k2 Hog1P
0  k3 Hog1P
In the steady-state:
OG
Hog1P  0
1. In the steady state Hog1P=0 independent from the outer
osmolarity.
 The system tracks the ‘desired’ steady state of Hog1P=0
independent from perturbations.
 Integral feedback property.
 ‘Perfect adaptor’ (in Hog1P).
2. Internal glycerol concentration depends on the outer osmolarity.
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A block diagram of integral feedback control.
• Variable u is the input for a process
with gain k.
• Difference between the actual
output y1 and the steady-state, i.e.
‘desired’, output y0 represents the
normalized output or error y.
• Integral control arises through the
feedback loop in which the time
integral of y, x, is fed back into the
system.
• As a result, we have ẋ = y and y = 0
at steady-state for all u.
Yi T et al. PNAS 2000;97:4649-4653
©2000ICYSB,
by National
Academy of Sciences
2013
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The mathematical model
• Variable u is the input for a
OuterOsmolarity
process with gain k.
time integral of y • Difference between the actual
output y1 and the steady-state,
Glycerol
Signal
i.e. ‘desired’, output y0
represents the normalized
output or error, y.
v
3
v1
• Integral control arises through
Hog1
P-Hog1
the feedback loop in which the
v2
time integral of y, x, is fed
y1
y0=0
back into the system.
y1=y
• As a result, we have ẋ = y and
y = 0 at steady-state for all u.
u
dHog1P
 k1 (O  G )(1  Hog1P )  k 2 Hog1P
dt
dG
 k3 Hog1P
dt
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Validation of the model
Single shock
Double shock
1 M KCl
0.5 M KCl
0.25 M KCl
- Simulations show reasonable fits and perfect adaptation
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Prediction
0.4 M KCl tripple shock (0, 30, 60 min)
P-Hog1l
InnerOsmolarity
OuterOsmolarity
- Simple model reacts to triple shock and shows perfect
adaptation.
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Remarks
Integral Feedback (3 parameters)
Auto-inhibitory feedback (20 parameters)
1 M KCl
0.5 M KCl
0.25 M KCl
r1
r2
r3
• From the quality of the fit, the auto-inhibitory
feedback model is much better than the integral
feedback model.
• The quality of the fit is usually measured by the sum
n
n
of squared residuals.
2
SSR( p, y )   ( yi  f i (t , p)    ri 2
i 1
i 1
• If it weren’t for the prediction, which model is better?
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Remarks
Integral Feedback (3 parameters)
Transient feedback (20 parameters)
1 M KCl
0.5 M KCl
0.25 M KCl
• Intuitively: both model capture the basic features of the data.
• In other words: The main information in the data is reproduced by
the models.
• But: the information/parameter is much higher in the simpler
model.
=> The parameters of the simpler model are more informative.
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The principle of parsimony
• If we take the number of parameters as a measure for structural
properties of the data, we want to have few parameters, i.e. the
most important structural features, and a good data representation.
• We do not want to have additional parameter to ‘fit the errors’ or spurious
effects.
• The simpler the model, the easier to analyse.
• Therefore, it is advisable to have a model that is as simple as possible
and as complex as necessary: this is the principle of parsimony.
“Everything should be made
as simple as possible, but not simpler.”
• Parsimony can be measured by the Akaike Information Criterion (AIC).

AIC  2k  n log SSR
n
• the lower AIC, the better the model approximates the data in terms of
parsimony (k number of parameters, n number of data).
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Model Selection
• The AIC can be used as a model selection criterion
# parameters
SSR
AIC
Transient FB
20
0.049
164.842
Integral FB
3
0.251
-38.045
• Only from the fits, the simper model would have been selected as
the best approximating model.
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Conclusions concerning feedback in
Sho1-branch
• According to the AIC, the data does not support a
model with auto-inhibitory feedback, but rather a model
with integral feedback.
• Therefore, for the adaptation and attenuation process,
the proposed feedback of Hog1 on Sho1 is not
necessary.
• The proposed feedback of Hog1 to Sho1 may modulate
the signal and serve other purposes (crosstalk,
stabilisation), but cannot explain adaptation to single and
multiple shock.
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Final Remarks on Sho1-Modelling
• The integral feedback model has several shortcoming:
• glycerol is only accumulated, never lost.
 Adaptation to a lowering in external osmolarity cannot be
modelled
• initial Hog1P is zero
All models are wrong, but some are useful.
• Our model was developed to address the question, whether or not
the (multiple) osmotic shock data can be explained by a transient
feedback, nothing else.
• All models are tailored to address specific questions. No model
can explain everything.
• The clearer our hypotheses are formulated, the better models can
be developed to address these.
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Final Remarks on Modelling
• The ‘truth’ (full reality) in biological sciences has infinite complexity
and, hence, can never be revealed with only finite samples and a
‘model’ of those data.
• It is a mistake to believe that there is a simple “true model” and
that during data analysis this model can be uncovered and its
parameters estimated.
• We can only hope to identify a model that provides a good
approximation to the data available.
• Uncertainty about the biology leads to multiple hypotheses about the
underlying processes explaining a set of data.
• I recommend the formulation of multiple working hypotheses, and the
building of a small set of models to clearly and uniquely represent these
hypotheses.
• The best approximating model can then be identified by model
selection criteria.
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