thermodynamic_modeling_of_NP_endocytosis

This repository contains the Python implementation of a thermodynamic model to evaluate how nanoparticle (NP) size influences membrane wrapping during endocytosis, as described in our research.

Overview

The model is built upon a free-energy framework incorporating:

• Mixing entropy of NP mixtures
• Ligand-receptor binding energy
• Membrane bending energy
• NP configurational entropy (simplified to retain only size-dependent energetic and entropic terms)

The free-energy functional used in the simulation is:
$$W = M_f[\xi_f \ln(\xi_f) + (1-\xi_f)\ln(1-\xi_f)] - \mu L_b + 8\pi\kappa\eta N$$

Where all energies are expressed in units of $k_BT$:

• $M_f$: Free membrane area (in units of receptor area $A_0$), $M_f = M - M_b$ (total membrane $M = 4\pi R^2/A_0$)
• $\xi_f$: Density of free receptors, $\xi_f = (\xi_0M - L_b)/M_f$
• $\mu$: Chemical energy gain from ligand-receptor binding
• $L_b = \eta NK$: Number of ligand-receptor bonds (scaled linearly with wrapped area)
• $\kappa$: Membrane bending rigidity (varied for clathrin-independent (CIE, $\kappa=10k_BT$) and clathrin-mediated (CME, $\kappa=60k_BT$) endocytosis)
• $\eta \in [0,1]$: Wrapping area fraction of NPs
• $N$: Number of NPs with wrapping fraction $\eta$ (contributes $8\pi\kappa\eta$ to total bending energy)
• $K = 4\pi R^2/A_0$: NP surface area in units of receptor area $A_0$
• $\xi_0$: Initial receptor density on the unbound membrane

For each NP size, the free energy $W$ is numerically minimized over $N$ (number of NPs) and $\eta$ (wrapping fraction). The resulting wrapping amount $\eta N$ is used as a proxy for endocytosis capacity.

Requirements

• Python 3.12+ (consistent with the research implementation)
• numpy >= 1.21.0
• matplotlib >= 3.4.0
• pandas >= 1.3.0

Install dependencies via pip:

pip install numpy matplotlib pandas

## How to Run
1. Clone this repository:
```bash
git clone https://github.com/[Your-Username]/[Repo-Name].git
cd [Repo-Name]

2. Execute the main simulation script:

python np_endocytosis_simulation.py

3. The script will: Run numerical minimization of free energy for NP sizes from 10 nm to 100 nm diameter Generate separate plots for CIE and CME pathways (y-axis tick labels hidden as per visualization requirements) Print optimal NP diameter (max endocytosis capacity) for both pathways in the terminal

Key Parameters

Parameter	Description	Unit	CIE Value	CME Value
$\kappa$ (kappa)	Membrane bending rigidity	$k_BT$	10	60
$\mu$ (mu)	Ligand-receptor binding energy gain	$k_BT$	20	20
$\xi_0$ (xi0)	Initial receptor density on unbound membrane	Dimensionless	0.05	0.05
$A_0$ (A0)	Area per receptor	$m^2$	$(15e-9)^2$	$(15e-9)^2$
$M$	Total membrane lattice points	Dimensionless	$3.14e6$	$3.14e6$
$c$	NP surface concentration	$1/A_0$	0.003	0.003

Implementation Details

• A grid-search approach is used to numerically minimize the free energy over N (number of NPs) and η (wrapping fraction)
• Physical constraints are enforced (e.g., non-negative free membrane area, valid receptor density range)
• Separate simulations for CIE and CME pathways (differentiated by bending rigidity κ)
• Output plots show NP diameter (x-axis) vs. endocytosis capacity (ηN, y-axis) with optimal diameter marked

Output

1. Interactive Plots:

• Two separate figures for CIE and CME endocytosis
• X-axis: NP diameter (nm)
• Y-axis: Wrapped/endocytosed amount (ηN)
• Vertical dashed line: Optimal NP diameter for maximum endocytosis capacity

2. Terminal Summary:

• Optimal diameter and maximum wrapped amount for CIE/CME pathways