# Interactions¶

Interactions are used to calculate forces on individual particles due to their neighbours. Pairwise short-range interactions are currently supported, and membrane forces.

## Summary¶

 Interaction() Base interaction class MembraneForces() Abstract class for membrane interactions. ObjRodBinding() Forces attaching a RodVector to a RigidObjectVector. Pairwise() Generic pairwise interaction class. RodForces() Forces acting on an elastic rod.

## Details¶

class Interaction

Bases: object

Base interaction class

__init__()

Initialize self. See help(type(self)) for accurate signature.

class MembraneForces

Abstract class for membrane interactions. Mesh-based forces acting on a membrane according to the model in [Fedosov2010]

The membrane interactions are composed of forces comming from:
• bending of the membrane, potential $$U_b$$
• shear elasticity of the membrane, potential $$U_s$$
• constraint: area conservation of the membrane (local and global), potential $$U_A$$
• constraint: volume of the cell (assuming incompressible fluid), potential $$U_V$$
• membrane viscosity, pairwise force $$\mathbf{F}^v$$
• membrane fluctuations, pairwise force $$\mathbf{F}^R$$

The form of the constraint potentials are given by (see [Fedosov2010] for more explanations):

$\begin{split}U_A = \frac{k_a (A_{tot} - A^0_{tot})^2}{2 A^0_{tot}} + \sum_{j \in {1 ... N_t}} \frac{k_d (A_j-A_0)^2}{2A_0}, \\ U_V = \frac{k_v (V-V^0_{tot})^2}{2 V^0_{tot}}.\end{split}$

The viscous and dissipation forces are central forces and are the same as DPD interactions with $$w(r) = 1$$ (no cutoff radius, applied to each bond).

Several bending models are implemented. First, the Kantor enrgy reads (see [kantor1987]):

$U_b = \sum_{j \in {1 ... N_s}} k_b \left[ 1-\cos(\theta_j - \theta_0) \right].$

The Juelicher energy is (see [Juelicher1996]):

$\begin{split}U_b = 2 k_b \sum_{\alpha = 1}^{N_v} \frac {\left( M_{\alpha} - C_0\right)^2}{A_\alpha}, \\ M_{\alpha} = \frac 1 4 \sum_{<i,j>}^{(\alpha)} l_{ij} \theta_{ij}.\end{split}$

It is improved with the ADE model (TODO: ref).

Currently, the stretching and shear energy models are:

WLC model:

$U_s = \sum_{j \in {1 ... N_s}} \left[ \frac {k_s l_m \left( 3x_j^2 - 2x_j^3 \right)}{4(1-x_j)} + \frac{k_p}{l_0} \right].$

Lim model: an extension of the Skalak shear energy (see [Lim2008]).

$\begin{split}U_{Lim} =& \sum_{i=1}^{N_{t}}\left(A_{0}\right)_{i}\left(\frac{k_a}{2}\left(\alpha_{i}^{2}+a_{3} \alpha_{i}^{3}+a_{4} \alpha_{i}^{4}\right)\right.\\ & +\mu\left(\beta_{i}+b_{1} \alpha_{i} \beta_{i}+b_{2} \beta_{i}^{2}\right) ),\end{split}$

where $$\alpha$$ and $$\beta$$ are the invariants of the strains.

 [Fedosov2010] (1, 2) Fedosov, D. A.; Caswell, B. & Karniadakis, G. E. A multiscale red blood cell model with accurate mechanics, rheology, and dynamics Biophysical journal, Elsevier, 2010, 98, 2215-2225
 [kantor1987] Kantor, Y. & Nelson, D. R. Phase transitions in flexible polymeric surfaces Physical Review A, APS, 1987, 36, 4020
 [Juelicher1996] Juelicher, Frank, and Reinhard Lipowsky. Shape transformations of vesicles with intramembrane domains. Physical Review E 53.3 (1996): 2670.
 [Lim2008] Lim HW, Gerald, Michael Wortis, and Ranjan Mukhopadhyay. Red blood cell shapes and shape transformations: newtonian mechanics of a composite membrane: sections 2.1–2.4. Soft Matter: Lipid Bilayers and Red Blood Cells 4 (2008): 83-139.
__init__(name: str, shear_desc: str, bending_desc: str, stress_free: bool = False, grow_until: float = 0.0, **kwargs) → None
Parameters: name – name of the interaction shear_desc – a string describing what shear force is used bending_desc – a string describing what bending force is used stress_free – if True, stress Free shape is used for the shear parameters grow_until – the size increases linearly in time from half of the provided mesh to its full size after that time; the parameters are scaled accordingly with time

kwargs:

• tot_area: total area of the membrane at equilibrium
• tot_volume: total volume of the membrane at equilibrium
• ka_tot: constraint energy for total area
• kv_tot: constraint energy for total volume
• kBT: fluctuation temperature (set to zero will switch off fluctuation forces)
• gammaC: central component of dissipative forces
• gammaT: tangential component of dissipative forces (warning: if non zero, the interaction will NOT conserve angular momentum)

Shear Parameters, warm like chain model (set shear_desc = ‘wlc’):

• x0: $$x_0$$
• ks: energy magnitude for bonds
• mpow: $$m$$
• ka: energy magnitude for local area

Shear Parameters, Lim model (set shear_desc = ‘Lim’):

• ka: $$k_a$$, magnitude of stretching force
• mu: $$\mu$$, magnitude of shear force
• a3: $$a_3$$, non linear part for stretching
• a4: $$a_4$$, non linear part for stretching
• b1: $$b_1$$, non linear part for shear
• b2: $$b_2$$, non linear part for shear

Bending Parameters, Kantor model (set bending_desc = ‘Kantor’):

• kb: local bending energy magnitude
• theta: spontaneous angle

Bending Parameters, Juelicher model (set bending_desc = ‘Juelicher’):

• kb: local bending energy magnitude
• C0: spontaneous curvature
• kad: area difference energy magnitude
• DA0: area difference at relaxed state divided by the offset of the leaflet midplanes
class ObjRodBinding

Forces attaching a RodVector to a RigidObjectVector.

__init__(name: str, torque: float, rel_anchor: float3, k_bound: float) → None
Parameters: name – name of the interaction torque – torque magnitude to apply to the rod rel_anchor – position of the anchor relative to the rigid object k_bound – anchor harmonic potential magnitude
class Pairwise

Generic pairwise interaction class. Can be applied between any kind of ParticleVector classes. The following interactions are currently implemented:

• DPD:

Pairwise interaction with conservative part and dissipative + random part acting as a thermostat, see [Groot1997]

$\begin{split}\mathbf{F}_{ij} &= \mathbf{F}^C(\mathbf{r}_{ij}) + \mathbf{F}^D(\mathbf{r}_{ij}, \mathbf{u}_{ij}) + \mathbf{F}^R(\mathbf{r}_{ij}) \\ \mathbf{F}^C(\mathbf{r}) &= \begin{cases} a(1-\frac{r}{r_c}) \mathbf{\hat r}, & r < r_c \\ 0, & r \geqslant r_c \end{cases} \\ \mathbf{F}^D(\mathbf{r}, \mathbf{u}) &= \gamma w^2(\frac{r}{r_c}) (\mathbf{r} \cdot \mathbf{u}) \mathbf{\hat r} \\ \mathbf{F}^R(\mathbf{r}) &= \sigma w(\frac{r}{r_c}) \, \theta \sqrt{\Delta t} \, \mathbf{\hat r}\end{split}$

where bold symbol means a vector, its regular counterpart means vector length: $$x = \left\lVert \mathbf{x} \right\rVert$$, hat-ed symbol is the normalized vector: $$\mathbf{\hat x} = \mathbf{x} / \left\lVert \mathbf{x} \right\rVert$$. Moreover, $$\theta$$ is the random variable with zero mean and unit variance, that is distributed independently of the interacting pair i-j, dissipation and random forces are related by the fluctuation-dissipation theorem: $$\sigma^2 = 2 \gamma \, k_B T$$; and $$w(r)$$ is the weight function that we define as follows:

$\begin{split}w(r) = \begin{cases} (1-r)^{p}, & r < 1 \\ 0, & r \geqslant 1 \end{cases}\end{split}$
• MDPD:

Compute MDPD interaction as described in [Warren2003]. Must be used together with “Density” interaction with kernel “MDPD”.

The interaction forces are the same as described in “DPD” with the modified conservative term

$F^C_{ij} = a w_c(r_{ij}) + b (\rho_i + \rho_j) w_d(r_{ij}),$

where $$\rho_i$$ is computed from “Density” and

$\begin{split}w_c(r) = \begin{cases} (1-\frac{r}{r_c}), & r < r_c \\ 0, & r \geqslant r_c \end{cases} \\ w_d(r) = \begin{cases} (1-\frac{r}{r_d}), & r < r_d \\ 0, & r \geqslant r_d \end{cases}\end{split}$
• SDPD:

Compute SDPD interaction with angular momentum conservation. Must be used together with “Density” interaction with the same density kernel.

The available density kernels are listed in “Density”. The available equations of state (EOS) are:

Linear equation of state:

$p(\rho) = c_S^2 \left(\rho - \rho_0 \right)$

where $$c_S$$ is the speed of sound and $$\rho_0$$ is a parameter.

Quasi incompressible EOS:

$p(\rho) = p_0 \left[ \left( \frac {\rho}{\rho_r} \right)^\gamma - 1 \right],$

where $$p_0$$, $$\rho_r$$ and $$\gamma = 7$$ are parameters to be fitted to the desired fluid.

• RepulsiveLJ:

Pairwise interaction according to the classical Lennard-Jones potential The force however is truncated such that it is always repulsive.

$\mathbf{F}_{ij} = \max \left[ 0.0, 24 \epsilon \left( 2\left( \frac{\sigma}{r_{ij}} \right)^{14} - \left( \frac{\sigma}{r_{ij}} \right)^{8} \right) \right]$

Note that in the implementation, the force is bounded for stability at larger time steps.

• Density:

Compute density of particles with a given kernel.

$\rho_i = \sum\limits_{j \neq i} w_\rho (r_{ij})$

where the summation goes over the neighbours of particle $$i$$ within a cutoff range of $$r_c$$. The implemented densities are listed below:

• kernel “MDPD”:

see [Warren2003]

$\begin{split}w_\rho(r) = \begin{cases} \frac{15}{2\pi r_d^3}\left(1-\frac{r}{r_d}\right)^2, & r < r_d \\ 0, & r \geqslant r_d \end{cases}\end{split}$
• kernel “WendlandC2”:

$w_\rho(r) = \frac{21}{2\pi} \left( 1 - \frac{r}{r_c} \right)^4 \left( 1 + 4 \frac{r}{r_c} \right)$
 [Groot1997] Groot, R. D., & Warren, P. B. (1997). Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulations. J. Chem. Phys., 107(11), 4423-4435. doi
 [Warren2003] Warren, P. B. “Vapor-liquid coexistence in many-body dissipative particle dynamics.” Physical Review E 68.6 (2003): 066702._
__init__(name: str, rc: float, kind: str, **kwargs) → None
Parameters: name – name of the interaction rc – interaction cut-off (no forces between particles further than rc apart) kind – interaction kind (e.g. DPD). See below for all possibilities.

Create one pairwise interaction handler of kind kind. When applicable, stress computation is activated by passing stress = True. This activates virial stress computation every stress_period time units (also passed in kwars)

• kind = “DPD”

• a: $$a$$
• gamma: $$\gamma$$
• kBT: $$k_B T$$
• power: $$p$$ in the weight function
• kind = “MDPD”

• rd: $$r_d$$
• a: $$a$$
• b: $$b$$
• gamma: $$\gamma$$
• kBT: temperature $$k_B T$$
• power: $$p$$ in the weight function
• kind = “SDPD”

• viscosity: fluid viscosity
• kBT: temperature $$k_B T$$
• EOS: the desired equation of state (see below)
• density_kernel: the desired density kernel (see below)
• kind = “RepulsiveLJ”

• epsilon: $$\varepsilon$$
• sigma: $$\sigma$$
• max_force: force magnitude will be capped to not exceed max_force
• aware_mode:
• if “None”, all particles interact with each other.
• if “Object”, the particles belonging to the same object in an object vector do not interact with each other. That restriction only applies if both Particle Vectors in the interactions are the same and is actually an Object Vector.
• if “Rod”, the particles interact with all other particles except with the ones which are below a given a distance (in number of segment) of the same rod vector. The distance is specified by the kwargs parameter min_segments_distance.
• kind = “Density”

• density_kernel: the desired density kernel (see below)

The available density kernels are “MDPD” and “WendlandC2”. Note that “MDPD” can not be used with SDPD interactions. MDPD interactions can use only “MDPD” density kernel.

For SDPD, the available equation of states are given below:

• EOS = “Linear” parameters:

• sound_speed: the speed of sound
• rho_0: background pressure in $$c_S$$ units
• EOS = “QuasiIncompressible” parameters:

• p0: $$p_0$$
• rho_r: $$\rho_r$$
setSpecificPair(pv1: ParticleVectors.ParticleVector, pv2: ParticleVectors.ParticleVector, **kwargs) → None

Set specific parameters of a given interaction for a specific pair of ParticleVector. This is useful when interactions only slightly differ between different pairs of ParticleVector. The specific parameters should be set in the kwargs field, with same naming as in construction of the interaction. Note that only the values of the parameters can be modified, not the kernel types (e.g. change of density kernel is not supported in the case of SDPD interactions).

Parameters: pv1 – first ParticleVector pv2 – second ParticleVector
class RodForces

Forces acting on an elastic rod.

The rod interactions are composed of forces comming from:
• bending energy, $$E_{\text{bend}}$$
• twist energy, $$E_{\text{twist}}$$
• bounds energy, $$E_{\text{bound}}$$

The form of the bending energy is given by (for a bi-segment):

$E_{\mathrm{bend}}=\frac{l}{4} \sum_{j=0}^{1}\left(\kappa^{j}-\overline{\kappa}\right)^{T} B\left(\kappa^{j}-\overline{\kappa}\right),$

where

$\kappa^{j}=\frac {1} {l} \left((\kappa \mathbf{b}) \cdot \mathbf{m}_{2}^{j},-(\kappa \mathbf{b}) \cdot \mathbf{m}_{1}^{j}\right).$

See, e.g. [bergou2008] for more details. The form of the twist energy is given by (for a bi-segment):

$E_{\mathrm{twist}}=\frac{k_{t} l}{2}\left(\frac{\theta^{1}-\theta^{0}}{l}-\overline{\tau}\right)^{2}.$

The additional bound energy is a simple harmonic potential with a given equilibrium length.

 [bergou2008] Bergou, M.; Wardetzky, M.; Robinson, S.; Audoly, B. & Grinspun, E. Discrete elastic rods ACM transactions on graphics (TOG), 2008, 27, 63
__init__`(name: str, state_update: str = 'none', save_energies: bool = False, **kwargs) → None
Parameters: name – name of the interaction state_update – description of the state update method; only makes sense for multiple states. See below for possible choices. save_energies – if True, save the energies of each bisegment

kwargs:

• a0 (float): equilibrium length between 2 opposite cross vertices
• l0 (float): equilibrium length between 2 consecutive vertices on the centerline
• k_s_center (float): elastic force magnitude for centerline
• k_s_frame (float): elastic force magnitude for material frame particles
• k_bending (float3): Bending symmetric tensor $$B$$ in the order $$\left(B_{xx}, B_{xy}, B_{zz} \right)$$
• kappa0 (float2): Spontaneous curvatures along the two material frames $$\overline{\kappa}$$
• k_twist (float): Twist energy magnitude $$k_\mathrm{twist}$$
• tau0 (float): Spontaneous twist $$\overline{\tau}$$
• E0 (float): (optional) energy ground state

state update parameters, for state_update = ‘smoothing’:

(not fully implemented yet; for now just takes minimum state but no smoothing term)

state update parameters, for state_update = ‘spin’:

• nsteps number of MC step per iteration
• kBT temperature used in the acceptance-rejection algorithm
• J neighbouring spin ‘dislike’ energy

The interaction can support multiple polymorphic states if kappa0, tau0 and E0 are lists of equal size. In this case, the E0 parameter is required. Only lists of 1, 2 and 11 states are supported.