An An It Iterat erative ive Par Parameter ameter Est Estim - - PowerPoint PPT Presentation

An An It Iterat erative ive Par Parameter ameter Est Estim imation ation Me Metho thod d fo for r Bio Biological logical Sy Syste stems ms

Xian Yang, Yike Guo, Jeremy Bradley

Overview

• Background
• Problem Formulation
• Parameter Estimation Methods
• Experiments and Results
• Discussion
• Conclusion

Background

• 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

• 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

• Background
• Problem Formulation
• Parameter Estimation Methods
• Experiments and Results
• Discussion
• Conclusion

Problem Formulation

• The biological process modelled by a system of differential equations is
• f 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

• 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

• A simple example:
• Model equations:

Where

Overview

• 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

• Parameter inference methods combine experimental data with the

knowledge about the underlying structure of a dynamical system to be investigated.

• 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.

• 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

• Advantage:
• They can infer the whole probability distributions of the parameters, rather

than just a point estimate.

• 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

• 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:

θ1 θ2 t X(t)

Parameter Estimation methods – Bayesian Methods

θ1 θ2 t X(t)

Parameter Estimation methods – Bayesian Methods

θ1 θ2 t X(t)

Parameter Estimation methods – Bayesian Methods

θ1 θ2 t X(t)

Parameter Estimation methods – Bayesian Methods

θ1 θ2 t X(t)

Parameter Estimation methods – Bayesian Methods

θ1 θ2 t X(t)

Parameter Estimation methods – Bayesian Methods

Parameter Estimation methods – Bayesian Methods

• Weakness of the ABC rejection sampler:
• High rejection rate
• 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

θ1 θ2

Population 1 ϵ1

θ1 θ2

Population 2 ϵ2

θ1 θ2

Population 3 ϵ3

Random sampling Perturbing

Parameter Estimation methods – In Inte tegrati gration

• n of
• f ABC

ABC SMC SMC an and wi d wind ndowing

• wing me

method thod

• 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 – In Inte tegrati gration

• n of
• f ABC

ABC SMC SMC an and wi d wind ndowing

• wing me

method thod

• 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 – In Inte tegrati gration

• n of
• f ABC

ABC SMC SMC an and wi d wind ndowing

• wing me

method thod

• The parameter estimation method proposed in this paper is the

combination of a windowing method and ABC SMC.

• 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 – In Inte tegrati gration

• n of
• f ABC

ABC SMC SMC an and wi d wind ndowing

• wing me

method thod

• 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

• 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 – Mod Model el of

• f a bi

a biologi

• logical

cal sy system stem

• 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

• 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.

5 10 15 20 25 30

• 10

10 20 30 40 50 60 70 80 Time(min) Concentrations (arbitrary units) measured mlacl measured mtetR measured mcl mlacl mtetR mcl

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 1000/148720=0.67%

0.5 1 1.5 2 50 100 150 200

0

1.5 2 2.5 50 100 150 200 n 3 4 5 6 50 100 150



500 1000 1500 2000 2500 50 100 150 200



.

• 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 {θ0

1 , θ0 2 ,…, θ0 1000 } 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,

where the acceptance rate is 1000/62216=1.6%.

.

0.5 1 1.5 2 50 100 150 200

0

1.5 2 2.5 50 100 150 200 n 3 4 5 6 50 100 150 200



500 1000 1500 2000 2500 50 100 150 200



Experiments and Results – Results of the ABC SMC method using equal sampling weight

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%.

.

0.5 1 1.5 2 50 100 150 200

0

1.5 2 2.5 50 100 150 200 n 3 4 5 6 50 100 150 200



500 1000 1500 2000 2500 50 100 150 200



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%.

.

0.5 1 1.5 2 50 100 150 200

0

1.5 2 2.5 50 100 150 200 n 3 4 5 6 50 100 150 200



500 1000 1500 2000 2500 50 100 150 200



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%.

.

0.5 1 1.5 2 50 100 150 200

0

1.5 2 2.5 50 100 150 200 n 3 4 5 6 50 100 150 200



500 1000 1500 2000 2500 50 100 150 200



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.

.

α0 n β α Mean value 1.128 2.0893 4.89 1011 Variance 0.001 0.0008 0.073 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

0.5 1 1.5 2 1.6 1.8 2 2.2 2.4 2.6

0 n

0.5 1 1.5 2 3 3.5 4 4.5 5 5.5 6 6.5

0 

0.5 1 1.5 2 500 1000 1500 2000 2500

0 

3 3.5 4 4.5 5 5.5 6 6.5 500 1000 1500 2000 2500

 

Error Allowance is 5000 Error Allowance is 1000 Error Allowance is 500 Error Allowance is 300 Error Allowance is 150

The outputs of ABC SMC as two-dimensional scatter plots.

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 Data generation steps Acceptance probability 5000 148720 0.67% 1000 33956 2.94% 500 29091 3.44% 300 48570 2.06% 150 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

The outputs of ABC SMC using adaptive sampling weight as two-dimensional scatter plots.

0.5 1 1.5 2 1.6 1.8 2 2.2 2.4 2.6

0 n

0.5 1 1.5 2 3 3.5 4 4.5 5 5.5 6 6.5

0 

0.5 1 1.5 2 500 1000 1500 2000 2500

0 

3 3.5 4 4.5 5 5.5 6 6.5 500 1000 1500 2000 2500

 

Error Allowance is 5000 Error Allowance is 1000 Error Allowance is 500 Error Allowance is 300 Error Allowance is 150

Overview

• Background
• Problem Formulation
• Parameter Estimation Methods
• Experiments and Results
• Discussion
• Conclusion

Discussion

• 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

5 10 15 20 25 30 10 20 30 40 50 60 70 80 Time(min) Concentrations Real mlacl Real mtetR Real mcl Simulated mlacl with 0 Simulated mtetR with 0 Simulated mcl with 0

• Figure. Compare the real dynamics of repressilator system with the simulated dynamics using parameter set .

Overview

• Background
• Problem Formulation
• Parameter Estimation Methods
• Experiments and Results
• Discussion
• Conclusion

Conclusion

• 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?