A cytoskeleton modeling method based on dissipative particle dynamics simulation

Abstract

The application provides a cytoskeleton modeling method based on dissipative particle dynamics simulation, comprising the following steps: step 1, setting a cell region in a dissipative particle dynamics simulation software; step 2, randomly distributing microfilament single chains which do not intersect each other in the cell region; step 3, randomly generating a plurality of free cross-linking protein particles in the cell region; step 4, the microfilament single chains and the cross-linking protein particles are polymerized / dissociated until the number of the cross-linking protein particles in three states tends to be stable, the three states of the cross-linking protein particles are free, combined with only one microfilament single chain and combined with two microfilament single chains; step 5, deleting the cross-linking protein particles and directly connecting two microfilament single chains connected by the same cross-linking protein particle; and step 6, calculating cytoskeleton parameters and outputting a cytoskeleton model. The cytoskeleton constructed by the application has high fidelity and presents mechanical characteristics such as viscoelasticity under the action of an external force.

Description

Technical Field

[0001] This invention belongs to the field of micromanipulation biological modeling technology, and specifically relates to a cytoskeleton modeling method based on dissipative particle dynamics simulation. Background Technology

[0002] Micromanipulation is a technique that involves manipulating cells or early embryos under a high-powered duplex microscope using micromanipulators. As an important manipulative method in cell engineering, micromanipulation is widely used in early embryo and cell manipulation. With the rapid development of computer and robotics technologies, robotic micromanipulation techniques have become increasingly sophisticated. Compared to manual manipulation, robotic micromanipulation does not require specialized personnel, has less contamination, higher repeatability, and a significantly improved success rate. However, during micromanipulation, the forces exerted on cells can damage them, affecting subsequent development and even leading to cell death. Therefore, improving cell survival rate and embryonic developmental potential after micromanipulation is crucial for the widespread adoption of this method. Current cell mechanics measurements require the assistance of sensors or exogenous markers, making it difficult to integrate measurement and manipulation, and sometimes even causing damage to the cells themselves. Therefore, finding suitable methods to explore the forces exerted on cells during micromanipulation is essential. Summary of the Invention

[0003] This invention addresses the technical problems existing in the prior art by providing a cytoskeleton modeling method based on dissipative particle dynamics simulation. The constructed cytoskeleton has high fidelity and exhibits mechanical properties such as viscoelasticity under external force.

[0004] The technical solution adopted in this invention is: a cytoskeleton modeling method based on dissipative particle dynamics simulation, comprising the following steps:

[0005] Step 1: Set non-periodic boundary conditions in the dissipative particle dynamics simulation software to determine the cell region;

[0006] Step 2: Establish a cytoskeleton microfilament in the dissipative particle dynamics simulation software. The cytoskeleton microfilament is a number of microfilament single chains that are randomly distributed in the cellular region and do not cross each other. Each microfilament single chain is a single chain model containing multiple microfilament protein particles. Randomly select a microfilament protein particle on each microfilament single chain as the binding site for cross-linking protein particles.

[0007] Step 3: Create cross-linked protein particles in dissipative particle dynamics simulation software. The cross-linked protein particles are single particles. Several free cross-linked protein particles are randomly generated in the cell region. Each cross-linked protein particle is bound to at most two single chains of the microfilament.

[0008] Step 4: The microfilament single chains are polymerized / dissociated with the cross-linked protein particles until the number of cross-linked protein particles in all three states tends to a steady state. The three states of the cross-linked protein particles are free, bound to only one microfilament single chain, and bound to two microfilament single chains.

[0009] Step 5: Delete the cross-linked protein particles and directly connect the binding sites of the two microfilament single chains to which the cross-linked protein particles are bound; that is, delete the free cross-linked protein particles and those bound to only one microfilament single chain directly without making any other changes; for cross-linked protein particles bound to two microfilament single chains, delete the cross-linked protein particles and connect the two microfilament single chains to which they are bound, and the sites where the two microfilament single chains are connected to each other are the original binding sites of each cross-linked protein particle;

[0010] Step 6: Calculate cytoskeleton parameters and output the cytoskeleton model.

[0011] Furthermore, in step 2, the microfilament single chain is a single-chain model containing 2-4 microfilament protein particles, with a length set to 3.5-4.5; the density of microfilament protein particles in the cellular region is set to 3.5;

[0012] In step 3, the density of cross-linked protein particles is set to 0.525.

[0013] Furthermore, in step 2, the number of microfilament protein particles on a single microfilament chain is determined based on its length, until the set density of microfilament protein particles is reached.

[0014] Furthermore, in step 3, periodic boundary conditions are set to obtain a particle generation region, which is located within the cell region, and the cross-linked protein particles are randomly generated within the particle generation region.

[0015] Furthermore, in step 4, the polymerization / depolymerization adopts the bonding / dissociation rate expression derived from the Boltzmann distribution based on affinity. When the cross-linked protein particles and the microfilament protein particles are close enough, they can bond at a rate k. on To form a bond, if the length of the existing bond exceeds the breaking distance, then at a rate k off fracture,

[0016]

[0017]

[0018] Where l is the instantaneous distance between the cross-linked protein particles and the microfilament protein particles, and l0 is the initial value. The bonding and dissociation rates, and the effective open strength σ, are respectively the values ​​when the distance between the microfilament protein particles and the cross-linked protein particles is l0.on and effective correlation strength σ off k represents the degree of decrease and increase in the corresponding rate within the interaction distance, respectively. B T is the unit of energy, k s is the elastic modulus of the bond.

[0019] Furthermore, in step 4, based on the dynamic rate, a probabilistic method is used to filter out unsuitable bonds in the simulation:

[0020]

[0021] Among them, P off d represents the probability of breaking a bond formed at a distance l. off It is the distance threshold.

[0022] Furthermore, in step 6, the cytoskeleton parameters are calculated by determining the two-body potential function U for the interaction between adjacent particles based on the material of the microfilament protein particles and cross-linked protein particles. bond Three-body potential function U angle and the four-body potential function U bending Parameters;

[0023] U bond =k filament (r-r0)

[0024] U angle =k angle (θ-θ0)

[0025]

[0026] Wherein, the k filamnet k is the spring constant. angle Let r be the bending stiffness, θ be the length of the spring, θ be the angle between two adjacent springs, r0 and θ0 be the initial values, and k be the bending stiffness. b It is the torsional constant, N d It is the number of angles formed by adjacent faces.

[0027] Compared with existing technologies, the beneficial effects of this invention are as follows: Based on dissipative particle dynamics, this invention randomly generates microfilament single chains and cross-linked protein particles in the cellular region, establishes a cross-linking mode between the microfilament single chains and cross-linked protein particles, incorporates the polymerization / depolymerization of microfilament single chains, and realizes the polymerization / depolymerization effect of the terminal particles of the microfilament single chains based on the kinetic rate. Under the action of external force, the cytoskeleton microfilament-cross-linked protein network exhibits viscoelastic mechanical properties. The simulation method of this invention simulates the cytoskeleton with high fidelity, which is of guiding significance for conducting cytoskeleton mechanical property analysis and micromanipulation optimization. Attached Figure Description

[0028] Figure 1This is a flowchart of an embodiment of the present invention;

[0029] Figure 2 This is a graph showing the number of cross-linked protein particles in three states during the polymerization / depolymerization process according to an embodiment of the present invention.

[0030] Figure 3 This is a cytoskeleton model established for an embodiment of the present invention. Detailed Implementation

[0031] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0032] Embodiments of the present invention provide a cytoskeleton modeling method based on dissipative particle dynamics simulation, such as... Figure 1 As shown, it includes the following steps:

[0033] Step 1: Set non-periodic boundary conditions in the dissipative particle dynamics simulation software to determine the cell region.

[0034] Step 2: Establish a cytoskeleton microfilament in dissipative particle dynamics simulation software. The cytoskeleton microfilament consists of several randomly distributed, non-intersecting single-chain microfilaments within the cellular region. Each single-chain microfilament contains multiple microfilament protein particles. A single microfilament protein particle is randomly selected from each single-chain microfilament as a binding site for cross-linked protein particles. The number of microfilament protein particles on each single-chain microfilament is determined based on its length, until a predetermined density of microfilament protein particles is reached. Each single-chain microfilament contains 2-4 microfilament protein particles and has a length set to 3.5-4.5 cm; the density of microfilament protein particles within the cellular region is set to 3.5.

[0035] Step 3: Create cross-linked protein particles in dissipative particle dynamics simulation software. Each cross-linked protein particle is a single particle, and each cross-linked protein particle is bound to at most two single chains of the microfilaments. Set periodic boundary conditions to obtain a particle generation region located within the cell region. The cross-linked protein particles are randomly generated within the particle generation region. The density of the cross-linked protein particles is set to 0.525.

[0036] Step 4: The microfilament single chains and cross-linked protein particles are polymerized / dissociated until the number of cross-linked protein particles in all three states tends to a steady state. The three states of cross-linked protein particles are free, bound to only one microfilament single chain, and bound to two microfilament single chains.

[0037] The polymerization / depolymerization process employs a bonding / dissociation rate expression derived from the Boltzmann distribution based on affinity. When the cross-linked protein particles and the microfilament protein particles are sufficiently close, they can bond at a rate k. onTo form a bond, if the length of the existing bond exceeds the breaking distance, then at a rate k off fracture,

[0038]

[0039]

[0040] Where l is the instantaneous distance between the cross-linked protein particles and the microfilament protein particles, and l0 is the initial value. The bonding and dissociation rates, and the effective open strength σ, are respectively the values ​​when the distance between the microfilament protein particles and the cross-linked protein particles is l0. on and effective correlation strength σ off k represents the degree of decrease and increase in the corresponding rate within the interaction distance, respectively. B T is the unit of energy, k s is the elastic modulus of the bond.

[0041] Based on dynamic rates, probabilistic methods are used to filter out unsuitable bonds in the simulation:

[0042]

[0043] Among them, P off d represents the probability of breaking a bond formed at a distance l. off It is the distance threshold.

[0044] In cytoskeleton modeling, the polymerization / depolymerization kinetics of the terminal particles of single-chain microfilaments are considered. This polymerization / depolymerization is determined by a probabilistic method. The rate of bonding and breaking is calculated based on the distance between two particles, and the probability in the above formula is calculated based on this rate. This probability determines which appropriate bonds should be removed, that is, to break the bonds between two particles and achieve depolymerization.

[0045] Step 5: Free cross-linked protein particles and those bound to only one microfilament single chain are directly deleted without any other changes; for cross-linked protein particles bound to two microfilament single chains, the cross-linked protein particle is deleted, and the two microfilament single chains bound to it are connected together. The sites where the two microfilament single chains are connected to each other are the original binding sites of each cross-linked protein particle.

[0046] Step 6: Calculate cytoskeleton parameters: Based on the material of the microfilament protein particles and cross-linked protein particles, determine the two-body potential function U for the interaction between adjacent particles. bond Three-body potential function U angle and the four-body potential function U bending Parameters;

[0047] U bond =k filament (r-r0)

[0048] U angle =k angle (θ-θ0)

[0049]

[0050] Wherein, the k filamnet k is the spring constant. angle Let r be the bending stiffness, θ be the length of the spring, θ be the angle between two adjacent springs, r0 and θ0 be the initial values, and k be the bending stiffness. b It is the torsional constant, N d It is the number of angles formed by adjacent faces.

[0051] Output the cytoskeleton model.

[0052] This embodiment employs the above method, using the dissipative particle dynamics simulation software LAMMPS (Large-scale Atomic / Molecular Massively Parallel Simulator) to simulate and model the cytoskeleton. LAMMPS is installed on an Ubuntu 16.04 Linux system, and OriginPro 2022b (Learning Edition) and OVITO Basic are used for visualization and data analysis. The specific process is as follows:

[0053] I. Generating Actin Microfilaments. An actin microfilament is composed of 2-4 actin particles linked together. The goal is to ensure that only one actin particle on each microfilament can bind to an actin crosslinking protein (ACP) particle, while maintaining that each microfilament is within the cell membrane. First, use the MATLAB program to generate actin microfilaments according to the following steps.

[0054] 1. The cytoskeleton must be located inside the cell membrane, and the area inside the cell membrane is the cellular region. Therefore, the generation range is set to a sphere with a radius of rc = 4, which is the selected range for generating actin particles. The selected density of generated actin particles is d_actin = 3.5, meaning that within this sphere, there will be a maximum of 3.5 protein particles per cubic unit on average.

[0055] 2. Randomly generate actin microfilaments within the cellular region. Taking an actin microfilament formed by four connected actin particles as an example, the following explains how to construct an actin microfilament. Randomly generate the 3D coordinates of an actin particle within the range rc. Randomly select a direction and simultaneously randomly generate a microfilament length within the range [3.9, 4.1], converting it into a vector increment. Add this vector to the coordinate vector of the generated particle to generate a pair of 3D coordinates in space, treating them as the actin particles at both ends of an actin microfilament. Divide the distance between the two actin particles generated in the previous step into three equal segments, treating the 3D coordinates of the endpoints of each segment as the 3D coordinates of an actin particle. We now obtain the 3D coordinates of the four actin particles on an actin microfilament. Use a fast rejection experiment algorithm to determine whether the actin microfilament intersects with other generated actin microfilaments. If it intersects, delete the actin microfilament and regenerate the actin microfilament starting from the random actin particle generation step. The binding sites of actin microfilaments and ACP were selected. Each actin microfilament has only one binding site for ACP, and the binding sites at different locations have different IDs.

[0056] 3. Repeat the above steps when the actin particle density is lower than that of d_actin.

[0057] 4. Print the data file in a Lammps-readable format, outputting the number and types of atoms, the number and types of bonds, the number and types of bond angles, the number and types of dihedral angles, bond IDs, and potential energy parameters, including the potential energy category: U. bond =k filament (r-r0), the spring constant k of this harmonic energy filamnet It is 8e5, the equilibrium length r0 is the Euclidean distance between the two atoms connected by the bond, the ID, type and three-dimensional coordinate position of each atom.

[0058] 2. Import the data file output from the MATLAB program into the Lammps software, add ACP, and polymerize the actin microfilaments and ACP. Actin cross-linked proteins are common intracellular proteins, and there are many types. Actin microfilaments form a cytoskeletal network by binding with ACP, and their morphology is often determined by the type of ACP. In this embodiment, we introduce only one type of ACP and connect it to a maximum of two different actin microfilaments, forming a cross-linked backbone through polymerization.

[0059] 1. Set the size of the Lammps simulation box. Due to the limitations of modern computer computing power, to approximate the simulation system, the boundary is set to a periodic boundary that can guarantee the conservation of particle number, momentum, and energy in the simulation. Obviously, this boundary value should be smaller than the radius of the cell membrane rc = 4. Therefore, the command `region cell sphere 0 0 0 3.9` is used to set a spherical region with a radius of 3.9 centered at (0, 0, 0).

[0060] 2. Set the above area as the simulation box. However, we still need to consider the generation of additional bonds and particles. Therefore, we need to set the number of atom types, the total number of bond types, the number of bond angle types, the number of additional generated bonds, and the number of bond angles. The command is: `create_box 3 cell bond / types 551 angle / types 2extra / bond / per / atom 10 extra / angle / per / atom 10 extra / special / per / atom 10`

[0061] 3. Set the potential function type of the atoms in the entire system, i.e., the dissipative particle dynamics type. In the simulation, the temperature is set to 1 and the cutoff radius is 2, i.e., pair_style dpd 1.0 2 1; in addition, the parameters of the conservative force, the dissipative force and the stage radius between particles should be set to 100, 65 and 0.5 respectively, i.e., pair_coeff**100 65 0.5.

[0062] 4. Use the command `create_atoms 1random 140 124132 cell` to generate 140 ACPs randomly within the simulated cell region. Additionally, use `read_data filename add append` to add actin microfilaments to the system.

[0063] 5. Since the polymerization / depolymerization process involves the formation / breakage of bonds between ACP and actin particles on actin microfilaments, it is necessary to set the potential energy type and parameters of this bond. The simulation sets the two-body potential between ACP and actin particles as a harmonic potential, with a spring constant of 8000, a balance length of 0.25, and the command `bond_style harmonic; bond_coeff 551 8000 0.25`. The angular potential energy between the three actin particles is also set as a harmonic potential, i.e., `angle_style harmonic`, with a bending stiffness k. angleThe value is 350, the equilibrium angle θ0 is 180, and the command is angle_coeff 1 350180; the three-body potential of actin particle-ACP-actin particle is set to harmonic potential, its bending stiffness constant is 550, and the equilibrium angle is 90, and the command is angle_coeff 2 550 90.

[0064] 6. Set the time step to timestep 0.0008.

[0065] 7. Use `minimize 1e-7 1e-7 100000 100000` to minimize energy and bring the system to a steady state. Then, use the `fix 1 all nve` command to add all particles to the `nve` system to ensure the energy conservation of the entire system.

[0066] 8. During the polymerization process, the single-bond polymerization and dissociation kinetics model proposed by Bell and the expressions for bonding and dissociation rates derived by Dembo et al. based on the affinity Boltzmann distribution were adopted, including the expression for bonding and dissociation rates under zero force. All values ​​are taken as 26e-4, and the switching strength σ is... off and σ on We take 3.5e-4, the polymerization distance l0 is 0.25, and the bond elasticity coefficient k s Take 8e4. Use the Lammps commands `fix 3 all bond / create 1 1 3 0.25 551 prob 1 238525 iparam 2 1 jparam 1 3` and `fix 4 all bond / break 50 551 0.25 prob 0.25 235825` to perform aggregation and deaggregation. To prevent particles from escaping the aggregation area, a spring wall is set in the simulation to bounce back particles attempting to fly out of the region cell. This requires the commands `region wall sphere 0 0 0 4.5` and `fix 2 all wall / region wall harmonic8000 0 0.5`.

[0067] 9. Configure the output file. Before using the dump command, you need to use the calculation command compute 1 allproperty / local batom1 batom2 btype to obtain the two atoms connected by the bond and the type of bond. Then, enter dump 1all custom 1000 all.lammpstrj id type xyz and dump 2 all custom 6000000end.lammpstrj id type xyz to output the particle state every 10,000 steps and the atom state at the end of the simulation, respectively, and print them to the file. At the same time, every 10,000 steps, the state of all bonds in the current system is output, including the bond ID, the IDs of the particles at both ends of the bond, and the type of bond. Finally, the state of the bonds in the system after the simulation is completed is output. These two commands are: dump 3 all local 10000 dump.bond.lammpstrj index c_1[1]c_1[2]c_1[3] and dump 4 all local 600000 dump.endb.lammpstrj index c_1[1]c_1[2]c_1[3].

[0068] 10. Set thermodynamic information output to output the current system temperature and bond state every 100,000 steps. The command is thermo 100000; thermo_style custom step temp bonds.

[0069] 11. The above process lasts for 6e5 time steps, i.e., run 600000. The number of ACP particles in all three states tends to a steady state, such as... Figure 2 As shown.

[0070] Third, trim the model to remove particles that have little impact on the calculation and improve computational efficiency.

[0071] 1. Use a MATLAB script to process the output file from the previous step, select the last frame after aggregation as the end, and use it as the state after the cytoskeleton aggregation is completed.

[0072] 2. Traverse the file and count the number of bonds in each step, the number of ACPs in the three states, and the number of free ACPs, ACPs bound to only one actin microfilament, and ACPs bound to two actin microfilaments.

[0073] 3. Remove free ACPs and ACPs that are bound to only one actin microfilament.

[0074] 4. For ACPs (Aspect-Channel Linked Particles) formed by two actin microfilaments, to replace the two cross-linked actin microfilament-ACP-actin microfilaments with actin microfilament-actin microfilament links, the script performs the following tasks: loads the file output and the output of the final bond step; reads the bond output; confirms whether the bond type is the bond connected to the ACP by judging the bond type; and counts the number of connections between each ACP and actin microfilaments. It reads the information of the generated microfilaments and checks the ACPs connected by two actin microfilaments. When checking the ACP connection, it is necessary to check the sites on the microfilaments connected to it. This is determined by reading the motor protein particle connection attributes in the generated actin microfilament file. At this point, the connections can be divided into three categories: both ACP binding sites are at one end of the two actin microfilaments; the ACP binding site is at one end of one actin microfilament and in the middle of the other actin microfilament; and the ACP binding site is in the middle of both actin microfilaments. Calculate the bond angles and dihedral angles for three different cases. Specifically: the number of dihedral angles varies depending on the binding sites; when both binding sites are at the ends, there is only one dihedral angle; when one binding site is at the end and the other in the middle, there are two dihedral angles; and when both binding sites are in the middle, there are four dihedral angles. Calculate the angle between the normal vectors of the dihedral angles, denoted as the equilibrium angle θ0 of the dihedral angle. At this time, the torsional constant k b Take 470.

[0075] 5. Output the modeling file to obtain the cytoskeleton model, such as... Figure 3 As shown.

[0076] The present invention has been described in detail above through embodiments, but the content described is only an exemplary embodiment of the present invention and should not be considered as limiting the scope of the present invention. The scope of protection of the present invention is defined by the claims. Any technical solutions designed by those skilled in the art using the technical solutions described in the present invention, or designed by those skilled in the art under the inspiration of the technical solutions of the present invention, within the substance and protection scope of the present invention, to achieve the above-mentioned technical effects, or any equivalent changes and improvements made to the scope of the application, should still fall within the patent protection scope of the present invention.

Claims

1. A method for cytoskeleton modeling based on dissipative particle dynamics simulation, characterized in that: Includes the following steps: Step 1: Set non-periodic boundary conditions in the dissipative particle dynamics simulation software to determine the cell region; Step 2: Establish a cytoskeleton microfilament in the dissipative particle dynamics simulation software. The cytoskeleton microfilament is a number of microfilament single chains that are randomly distributed in the cellular region and do not cross each other. Each microfilament single chain is a single chain model containing multiple microfilament protein particles. Randomly select a microfilament protein particle on each microfilament single chain as the binding site for cross-linking protein particles. Step 3: Create cross-linked protein particles in dissipative particle dynamics simulation software. The cross-linked protein particles are single particles. Several free cross-linked protein particles are randomly generated in the cell region. Each cross-linked protein particle is bound to at most two single chains of the microfilament. Step 4: The microfilament single chains are polymerized / dissociated with the cross-linked protein particles until the number of cross-linked protein particles in all three states tends to a steady state. The three states of the cross-linked protein particles are free, bound to only one microfilament single chain, and bound to two microfilament single chains. Step 5: Delete the cross-linked protein particles and directly connect the binding sites of the two microfilament single chains to which the cross-linked protein particles are bound; Step 6: Calculate cytoskeleton parameters and output the cytoskeleton model; In step 4, the polymerization / dissociation employs bond formation / dissociation rate expressions derived from the Boltzmann distribution of affinities, which can occur at a rate Formation of a bond, if the length of an existing bond exceeds the breaking distance, at a rate Breaking, ； in, This represents the instantaneous distance between cross-linked protein particles and microfilament protein particles. As the initial value, The distances between the microfilament protein particles and the cross-linked protein particles are respectively: The bonding and dissociation rates at that time, and the effective open strength and effective correlation strength These represent the degree of decrease and increase in the corresponding rate within the interaction distance, respectively. As a unit of energy, The elastic modulus of the bond; In step 4, based on the dynamic rate, a probabilistic method is used to filter out unsuitable bonds in the simulation: ; in, Indicates in The probability of breaking existing bonds at a given distance. It is the distance threshold.

2. The Dissipative Particle Dynamics Simulation based cytoskeleton modeling method of claim 1, wherein: In step 2, the single-chain microfilament is a single-chain model containing 2-4 microfilament protein particles, with a length set to 3.5-4.5; the density of microfilament protein particles in the cellular region is set to 3.

5. In step 3, the density of cross-linked protein particles is set to 0.

525.

3. The Dissipative Particle Dynamics Simulation-based cytoskeleton modeling method according to claim 1 or 2, characterized by: In step 2, the number of microfilament protein particles on a single microfilament chain is determined based on its length until the set density of microfilament protein particles is reached.

4. The Dissipative Particle Dynamics Simulation based cytoskeleton modeling method of claim 1, wherein: In step 3, periodic boundary conditions are set to obtain a particle generation region, which is located within the cell region, and the cross-linked protein particles are randomly generated within the particle generation region.

5. The Dissipative Particle Dynamics Simulation based cytoskeleton modeling method of claim 1, wherein: In step 6, parameters of the cytoskeletal parameters are calculated as parameters of two-body potential functions of interaction between adjacent particles, depending on the material of the microfilament protein particles and the cross-linking protein particles , three-body potential functions , and four-body potential functions . ； Among them, the Let be the spring constant. For bending stiffness, It is the length of the spring. It is the angle between two adjacent springs. and It is the initial value. It is the torsional constant. It is the number of angles formed by adjacent faces.

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