Parameter Estimation in Biological Modeling
A Case Study on Fitting a Logistic Growth Model Using Least Squares Estimation and Time-Series Data
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Parameter Estimation in Biological Modeling
Model parameters link abstract mathematical structures, or symbolic models, with the real-world biological data they aim to represent. Even the most sophisticated model cannot accurately reproduce, explain, or predict biological phenomena if its parameters are incorrect. Moreover, many analyses are impossible to conduct if the parameter values for a given model are unknown.
Parameter estimation is generally framed as an optimization problem. The objective is to find a set of parameter values (a parameter vector), each corresponding to a different model parameter, that minimizes the discrepancy between the model and experimental data. To avoid canceling out positive and negative differences in the data, an objective function is used. In most cases, this function is designed to minimize the sum of squared errors (SSE), which is the squared difference between experimental data and model predictions. The methods used to minimize SSE are known as least-squares methods1.
When a model is linear, or can be approximately transformed into a linear form, various efficient and well-established least-squares methods can be employed. However, when the model is nonlinear, these methods are often unsuitable. In such cases, we must resort to either (1) brute-force computation or (2) the development of specialized methods that are applicable to specific classes of nonlinear problems.
In some cases, it is possible to estimate parameters for individual processes within a system or model, and then combine these local estimates to form a comprehensive model that captures the interactions between processes. This bottom-up approach to parameter estimation has been successfully used in the past, but it often falls short. One key reason is that estimates for different parameters typically come from different experiments, conducted under varying conditions or with different organisms. Combining such disparate "local" information tends to result in a model that poorly approximates reality.
A more modern and effective approach to parameter estimation in large-scale biological systems is to use time-series data derived from genomic, proteomic, or metabolomic measurements. For instance, consider a biological system in its normal state. At the start of an experiment, we perturb the system and measure the responses of multiple components over time. The resulting time-series data holds valuable insights into the structure, regulation, and dynamics of the system, all collected from a single organism under consistent experimental conditions. However, extracting meaningful information from this time-series data is computationally challenging, and often requires significant creativity from researchers.
Case Study: Fitting a Simple Biological Model Using Least Squares Estimation
Let's apply parameter estimation to a simple model that describes the growth of a population. A commonly used model is the logistic growth equation:
Where P(t) is the population size at time t, P0 is the initial population size, K is the carrying capacity of the environment, r is the growth rate, and t is time.
In this example, we will use synthetic data for population growth, and attempt to estimate the parameters K, P0, and r by minimizing the sum of squared errors between the model and the data.
Step 1: Generate Synthetic Data
import numpy as np
import matplotlib.pyplot as plt
# True parameters
K_true = 1000
P0_true = 50
r_true = 0.1
t = np.linspace(0, 50, 100)
# Logistic growth model
def logistic_growth(t, P0, K, r):
return K / (1 + ((K - P0) / P0) * np.exp(-r * t))
# Generate synthetic population data with noise
population_data = logistic_growth(t, P0_true, K_true, r_true) + np.random.normal(0, 10, len(t))
# Plot the synthetic data
plt.scatter(t, population_data, label='Synthetic Data', color='blue')
plt.plot(t, logistic_growth(t, P0_true, K_true, r_true), label='True Growth Model', color='red')
plt.xlabel('Time')
plt.ylabel('Population Size')
plt.legend()
plt.show()Which produces the following output:
Step 2: Perform Parameter Estimation Using Least Squares
We will now use a least-squares approach to estimate the parameters K (the carrying capacity), P0 (the initial population), and r (the growth rate), as demonstrated below:
from scipy.optimize import curve_fit
# Define the model function for curve fitting
def logistic_model(t, P0, K, r):
return K / (1 + ((K - P0) / P0) * np.exp(-r * t))
# Fit the model to the data
params, covariance = curve_fit(logistic_model, t, population_data, p0=[50, 1000, 0.1])
# Extract the fitted parameters
P0_est, K_est, r_est = params
print(f"Estimated P0: {P0_est:.2f}")
print(f"Estimated K: {K_est:.2f}")
print(f"Estimated r: {r_est:.4f}")
# Plot the fitted model
plt.scatter(t, population_data, label='Synthetic Data', color='blue')
plt.plot(t, logistic_model(t, *params), label='Fitted Model', color='green')
plt.xlabel('Time')
plt.ylabel('Population Size')
plt.legend()
plt.show()After running the code above, you should see the estimated parameters printed in the console, along with a plot comparing the synthetic data with the fitted model, as demonstrated below:
The estimated parameters should be close to the true values used to generate the data, though there may be some error due to the noise added to the synthetic data. This same general process, demonstrated above, can be extended to more complex biological systems, such as gene regulatory networks or metabolic pathways, where time-series data can provide insights into system dynamics.
Note: SSE is also referred to as the "error function," so parameter estimation tasks are essentially equivalent to tasks of minimizing the error function.