An Iterative Parameter Estimation
Method for Biological Systems
Xian Yang, Yike Guo, Jeremy Bradley
Overview
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Background
Problem Formulation
Parameter Estimation Methods
Experiments and Results
Discussion
Conclusion
Background
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To interpret the time evolution of biological systems, scientists build up
models, which are their simplified mathematical representations.
These models are essentially a set of equations whose solution
describes the evolution, as a function of time, of the state of the
system.
They include the following ingredients:
• A phase space S
• Time t
• An evolution law
Example:
• dx/dt = X(x)
• xt+1 = X(xt)
Background
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Mechanistic models of pathways describe the time evolution of
molecules and give a detailed insight into pathway dynamics.
Some parameters, such as kinetic rates and initial concentrations,
cannot be measured directly by biological experiments.
It is necessary to estimate unknown parameters from the observed
system dynamics.
Given the specified structure of pathway and ranges of parameter
values to be explored, parameter estimation methods search the
parameter space to generate parameters for which the simulated
model can exhibit the desired behaviour.
Overview
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Background
Problem Formulation
Parameter Estimation Methods
Experiments and Results
Discussion
Conclusion
Problem Formulation
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The biological process modelled by a system of differential equations is
of the form:
where
contains the unknown parameters that we seek
to estimate,
represents the concentration
levels of M species involved in the system, such as proteins and
mRNA, at time t, and
shows the initial concentration levels of these
molecules.
Problem Formulation
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We represent quantities, , that can be measured experimentally as:
where h is the output function that relates the measurement with
system state. Usually, can only be measured at discrete time instant
where
and corrupted with random noise
.
Problem Formulation
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A simple example:
• Model equations:
Where
Overview
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Background
Problem Formulation
Parameter Estimation Methods
• Other methods
• Bayesian methods: ABC SMC
• Integration of ABC SMC and windowing method
Experiments and Results
Discussion
Conclusion
Parameter Estimation methods – Other methods
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Parameter inference methods combine experimental data with the
knowledge about the underlying structure of a dynamical system to be
investigated.
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Optimization methods:
• Formulate the parameter estimation problem as a nonlinear optimization
problem. Its objective function is the difference between model prediction and
the experimental data.
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Kalman filters:
• Parameter estimation is handled in the framework of control theory by using
state observations.
• They are recursive estimator that incorporates new information from
experimental data.
Parameter Estimation methods – Bayesian Methods
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Advantage:
• They can infer the whole probability distributions of the parameters, rather
than just a point estimate.
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Bayes’ rule for parameter estimation:
• Suppose we get the measurable quantities
, which reflect
the system state
, then the posterior distribution of
parameters is:
that is,
Parameter Estimation methods – Bayesian Methods
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Approximate Bayesian computation (ABC):
• For complex models, computing the likelihood
is time consuming and
intractable.
• ABC method avoid explicit evaluation of the likelihood by considering
distances between observvation and data simulated from a model with
parameter .
• The generic ABC approach to infer the posterior probability distribution of
is:
Parameter Estimation methods – Bayesian Methods
θ2
X(t)
θ1
t
Parameter Estimation methods – Bayesian Methods
θ2
X(t)
θ1
t
Parameter Estimation methods – Bayesian Methods
θ2
X(t)
θ1
t
Parameter Estimation methods – Bayesian Methods
θ2
X(t)
θ1
t
Parameter Estimation methods – Bayesian Methods
θ2
X(t)
θ1
t
Parameter Estimation methods – Bayesian Methods
θ2
X(t)
θ1
t
Parameter Estimation methods – Bayesian Methods
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Weakness of the ABC rejection sampler:
• High rejection rate
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ABC Sequential Monte Carlo (SMC) method:
• The ABC SMC method is used to obtain posterior distribution
via
intermediate distributions
, where
and P is the
total number of intermediate distributions. It should satisfy the condition that
ϵp+1< ϵp for all p.
• Note that the first step of the ABC SMC method is the ABC rejection
algorithm.
• For the special case, when the prior distribution of parameter and the
perturbation kernel are uniform, the sampling weight becomes 1/N.
Parameter Estimation methods – Bayesian Methods
θ2
Population 1
ϵ1
θ2
Population 2
ϵ2
θ2
Population 3
ϵ3
Perturbing
Random sampling
θ1
θ1
θ1
Parameter Estimation methods – Integration of ABC SMC
and windowing method
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When the prior knowledge of parameters is poor and the parameter
space to be explored is therefore large, the first run of the ABC SMC
method, which corresponds to the ABC rejection algorithm, will take a
long time to find the first intermediate distribution.
To solve this problem, we introduce a windowing method.
Parameter Estimation methods – Integration of ABC SMC
and windowing method
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Assume two parameters, θ(1) and θ(2), are to be estimated. The prior
distribution of these two parameters are Pr(θ(1))=U(ϴ1, ϴ2) and
Pr(θ(2))=U(ϴ3, ϴ4) the error allowance is ϵ.
Parameter Estimation methods – Integration of ABC SMC
and windowing method
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The parameter estimation method proposed in this paper is the
combination of a windowing method and ABC SMC.
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The windowing method is applied first to reduce the parameter space
to be explored by ABC SMC. The process of this integrated method is
as follows:
1. Using the windowing method to get a better prior knowledge of parameters.
2. By running the ABC SMC scheme, the target posterior distribution of
parameters is approached by a set of intermediate distributions.
Parameter Estimation methods – Integration of ABC SMC
and windowing method
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In this paper, the prior distribution of parameters is set to be uniform,
and the perturbation kernel in SMC is also uniform.
Therefore, the sampling weight equals to 1/N for all p and n.
The ABC SMC method can be improved if we adjust the weight value
corresponding to the distance.
Overview
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Background
Problem Formulation
Parameter Estimation Methods
Experiments and Results
• Model of a biological system
• Results of the windowing method
• Results of the ABC SMC method using equal sampling
weight
• Results of the ABC SMC method using adaptive
sampling weight
Discussion
Conclusion
Experiments and Results – Model of a biological system
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In this paper, we infer the parameters of a standard repressilator
model.
This model consists of six differential equations that represent the
concentration levels of three mRNA (where i ϵ {lacl, tetR, cl}) and three
repressor proteins (where j ϵ {cl, lacl, tetR}). The mathematical model
that represents this system is:
where
Experiments and Results – Settings of experiment
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Settings of experiment:
• Suppose we are able to measure the concentration level of mRNA at discrete
time instants
.
• The noise in the measurement is assumed to give a standard derivation of
5% of the mean of the signal.
• The initial concentrations of these six species are:
• The model is simulated by MATLAB’s ODE45 solver.
• In the simulation, the parameters that generate observations are:
• The distance between the noisy observation
and the solution of the system
for
is:
• The prior distribution of each parameter is assumed to be uniform that:
Experiments and Results – Settings of experiment
The oscillatory dynamics of the repressilator system with original
experimental settings.
measured mlacl
80
measured mtetR
measured mcl
70
mlacl
60
Concentrations (arbitrary units)
•
mtetR
mcl
50
40
30
20
10
0
-10
0
5
10
15
Time(min)
20
25
30
Experiments and Results – Results of the windowing
method
1. Results of the windowing method:
• Partition the whole large space into small regions whose central points are:
• We set the error threshold to be 35000 initially which is much larger than the
desired value.
• Use the performance of the central point of each region to represent the
whole region.
• 62 parameter points are selected (acceptance rate is 0.22%) with the
distance value smaller than 35000.
• The ranges of each parameter after windowing method are :
• After using the windowing method, the prior distributions of parameters which
are the inputs of the ABC SMC method are:
.
Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
• The error threshold is set successively to be
• The number of accepted parameter sets equals 1000.
• The first run of the ABC SMC method is an ABC rejection sampling scheme
with the error threshold ϵ0=5000.
• In order to get 1000 accepted parameter sets, 148720 parameter sets are
randomly sampled from prior distributions. Therefore, the acceptance rate is
200
200
1000/148720=0.67%
150
150
100
100
50
50
0
0
0.5
1
1.5
2
0
1.5
2
n
0
150
2.5
200
150
100
100
50
0
50
3
4
5

6
0
500
1000
1500

2000
2500
.
Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
• In the second iteration of the ABC SMC method, the parameter set is
sampled from the previous distribution {θ01 , θ02 ,…, θ01000 } with weight
1/1000.
• Then the sampled parameter set is perturbed with the uniform function
Kt=σU(-1,1), where σ =0.1 for α0,n,β and σ =0.5 for α.
• The perturbed parameter set is accepted if its distance value is less than
1000.
• To get 1000 parameter sets accepted, 62216 parameter
sets are checked,
200
200
where the acceptance rate is 1000/62216=1.6%. 150
150
100
100
50
50
0
0
0.5
1
1.5
2
0
1.5
2
n
0
200
200
150
150
100
100
50
50
0
3
4
5

6
0
500
1000
1500

2.5
2000
2500
.
Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
• In the third iteration of the ABC SMC method, same perturbation function is
used.
• The perturbed parameter set is accepted if its distance value is less than 500.
• The total number of parameter sets sampled from previous distribution is
43731 with an acceptance rate of 1000/43731=2.3%.
200
200
150
150
100
100
50
50
0
0
0.5
1
1.5
2
0
1.5
2
n
0
200
200
150
150
100
100
50
50
0
3
4
5

6
0
500
1000
1500

2.5
2000
2500
.
Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
• In the fourth iteration of the ABC SMC method, same perturbation function is
used.
• The perturbed parameter set is accepted if its distance value is less than 300.
• The total number of parameter sets sampled from previous distribution is
71594 with an acceptance rate of 1000/71594=1.4%.
200
200
150
150
100
100
50
50
0
0
0.5
1
1.5
2
0
1.5
2
n
0
200
200
150
150
100
100
50
50
0
3
4
5

6
0
500
1000
1500

2.5
2000
2500
.
Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
• In the fifth iteration of the ABC SMC method, same perturbation function is
used.
• The perturbed parameter set is accepted if its distance value is less than 150.
• The total number of parameter sets sampled from previous distribution is
582685 with an acceptance rate of 1000/ 582685 =0.17%.
200
200
150
150
100
100
50
50
0
0
0.5
1
1.5
2
0
1.5
2
n
0
200
200
150
150
100
100
50
50
0
3
4
5

6
0
500
1000
1500

2.5
2000
2500
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Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
Table 1. The number of data generation steps needed to accept 1000 parameter sets to
generate each intermediate distribution and its corresponding acceptance rate.
ϵ
Data generation steps
Acceptance probability
5000
148720
0.67%
1000
62216
1.6%
500
43731
2.3%
300
71594
1.4%
150
582685
0.17%
Total number of data generation steps = 908946
.
Experiments and Results – Results of the ABC SMC
method using equal sampling weight
2. Results of the ABC SMC method using equal sampling weight
Table 2. The mean and variance of each estimated parameter.
Mean value
Variance
α0
1.128
0.001
n
2.0893
0.0008
β
4.89
0.073
α
1011
1625
Experiments and Results – Results of the ABC SMC
method using adaptive sampling weight
2. Results of the ABC SMC method using equal sampling weight
Error
Error
Error
Error
Error
2.6
2.4
Allowance
Allowance
Allowance
Allowance
Allowance
is
is
is
is
is
5000
1000
500
300
150
6.5
6
5.5
2.2

n
5
4.5
2
4
1.8
3.5
0
0.5
1

1.5
3
2
2500
2500
2000
2000
1500
1500
1000
500
0
0.5
1

0


1.6
1.5
2
0
1000
0
0.5
1

0
1.5
2
500
3
3.5
4
4.5

The outputs of ABC SMC as two-dimensional scatter plots.
5
5.5
6
6.5
.
Experiments and Results – Results of the ABC SMC
method using adaptive sampling weight
3. Results of the ABC SMC method using adaptive sampling weight
Table 3. The number of data generation steps needed to accept 1000 parameter sets
to generate each intermediate distribution and its corresponding acceptance rate
using adaptive weight.
ϵ
5000
1000
500
300
150
Data generation steps
Acceptance probability
148720
0.67%
33956
2.94%
29091
3.44%
48570
2.06%
446048
0.22%
Total number of data generation steps = 706385
Experiments and Results – Results of the ABC SMC
method using adaptive sampling weight
3. Results of the ABC SMC method using adaptive sampling weight
Error
Error
Error
Error
Error
2.6
2.4
Allowance
Allowance
Allowance
Allowance
Allowance
is
is
is
is
is
5000
1000
500
300
150
6.5
6
5.5
2.2

n
5
4.5
2
4
1.8
3.5
0
0.5
1

1.5
3
2
2500
2500
2000
2000
1500
1500
1000
500
0
0.5
1

0


1.6
1.5
2
0
1000
0
0.5
1

0
1.5
2
500
3
3.5
4
4.5

5
5.5
6
6.5
The outputs of ABC SMC using adaptive sampling weight as two-dimensional scatter plots.
Overview
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Background
Problem Formulation
Parameter Estimation Methods
Experiments and Results
Discussion
Conclusion
Discussion
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We would like to compare our proposed parameter estimation method
with the methods proposed in [1].
It makes use of the probabilistic information in the measurement noise
and let parameter estimation be a nonlinear optimization problem.
The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is used in [1]
to search the desired parameters.
As with other optimization methods, BFGS can only return a single
parameter set rather than the posterior distributions of parameters.
Moreover, we find that the BFGS method may sometimes return a
parameter set whose value is far from the real one but its simulated
data is almost consistent with observations.
In our example, the BFGS based parameter estimation method
predicts parameters to have the following values:
[1] Lillacci, G. and Khammash, M. 2010. Parameter estimation and model selection in computational biology. PLoS computational biology, 6, 3 (March 2010), e1000696.
Discussion
80
Real mlacl
Real mtetR
Real mcl
70
Simulated m lacl with 0
Simulated m tetR with 0
Simulated m cl with 0
60
Concentrations
50
40
30
20
10
0
0
5
10
15
Time(min)
20
25
Figure. Compare the real dynamics of repressilator system with the simulated dynamics using parameter set .
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Overview
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Background
Problem Formulation
Parameter Estimation Methods
Experiments and Results
Discussion
Conclusion
Conclusion
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This paper develops an iterative parameter estimation method to
efficiently infer parameters of biological systems with a known
mathematical model, experimental measurements and ranges of
parameters.
Parameters in the mechanistic model of a repressilator system are
predicted by the proposed estimation method.
In order to increase the efficiency of the ABC SMC method, a
windowing technique is introduced to reduce the size of parameter
space to be explored.
Moreover, the ABC SMC method in this paper uses an adaptive
sampling weight which potentially reduces the number of data
generation steps.
Questions?