2.4 Rigid domains

Because a map object contains a large amount of data, it is inconvenient to perform movement on a map itself. For example, a rotation of a map object will result in the rectangular space not parallel to the coordinate axis anymore, and new boundaries and distributions need be updated accordingly. In addition, a real system often contains more than one copy of some molecular species, and it would be very memory costing to have a map object for each copy of these species. Instead, we define a rigid domain to represent a copy of the molecular species. A rigid domain contains only the identity of the map object it represents and the position and orientation vectors related to the map object, and can be manipulated easily. A rigid domain can be understood as a mobile representation of a map object. Each rigid domain has a unique identity, and many rigid domains can represent the same map object. Figure 2 shows the map objects of the α-chain and β-chain of a TCR variable domain and their manipulation through rigid domains.

Each rigid domain is defined by its map ID and its translation vector, T, and rotational matrix, U:

$$\mathbf{T} = \begin{pmatrix} t\_x \\ t\_y \\ t\_x \end{pmatrix}, \mathbf{U} = \begin{pmatrix} u\_{11} & u\_{12} & u\_{13} \\ u\_{21} & u\_{22} & u\_{23} \\ u\_{31} & u\_{32} & u\_{33} \end{pmatrix} \tag{3}$$

<sup>T</sup>ð Þ <sup>n</sup> <sup>¼</sup> <sup>T</sup>ð Þ <sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>Δ</sup>Tð Þ <sup>n</sup> <sup>¼</sup> <sup>T</sup>ð Þ <sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>Δ</sup>Tð Þ <sup>n</sup> <sup>þ</sup> <sup>Δ</sup>Tð Þ <sup>n</sup>�<sup>1</sup> <sup>¼</sup> <sup>T</sup><sup>0</sup>Þ þ <sup>∑</sup><sup>n</sup>

<sup>U</sup>ð Þ <sup>n</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>n</sup> � <sup>U</sup>ð Þ <sup>n</sup>�<sup>1</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>n</sup> � <sup>Ω</sup>ð Þ <sup>n</sup>�<sup>1</sup> � <sup>U</sup>ð Þ <sup>n</sup>�<sup>2</sup> <sup>¼</sup> <sup>Y</sup><sup>n</sup>

0 0 0 1

CA, <sup>U</sup> <sup>¼</sup>

A rigid domain can also be created for a molecular structure by linking the structure to a given map object according to the translation vector and rotation

0

B@

matrix from the reference coordinates to the linked structure.

to the position and orientation of a rigid domain.

between map objects, which are listed below.

1. Density correlation (DC)

2. Laplacian correlation (LC)

the following finite difference approximation:

2.5 Map comparison

where

and

67

T ¼

Protein-Protein Docking Using Map Objects DOI: http://dx.doi.org/10.5772/intechopen.83543

When a map object is created, a rigid domain at origin is created for it, with

0

B@

This equation also provides a way to update the structure coordinates according

Map comparison provides a target function for fitting one map into another map. Four types of cross-correlation functions [10] are provided for comparison

DCmn <sup>¼</sup> <sup>ρ</sup>mρ<sup>n</sup> � <sup>ρ</sup>mρ<sup>n</sup>

∑ nx i ∑ ny j ∑ nz k

q

represent the average and fluctuation of the density distribution. DCmn is the density correlation of map m to map n. Figure 3 shows two comparison maps in two dimensions. DCmnis calculated according to map m's dimension and grid properties. The calculation runs over all grid points of map m, which are transformed and interpolated into grid points of map n to get corresponding density properties.

<sup>ρ</sup>m∇<sup>2</sup>ρ<sup>n</sup> � <sup>∇</sup><sup>2</sup>

where ∇<sup>2</sup>ρ is the Laplacian filtered density derived from density distribution by

δ ∇<sup>2</sup> ρm � �δ ∇<sup>2</sup>

ρm∇<sup>2</sup>ρ<sup>n</sup>

ρn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ρ</sup><sup>2</sup> � <sup>ρ</sup><sup>2</sup>

<sup>ρ</sup> <sup>¼</sup> <sup>1</sup> nxnynz

LCmn <sup>¼</sup> <sup>∇</sup><sup>2</sup>

δ ρð Þ¼

100 010 001 1

<sup>X</sup> <sup>¼</sup> <sup>T</sup> <sup>þ</sup> <sup>U</sup> � <sup>X</sup>ð Þ ref (9)

δ ρð Þ <sup>m</sup> δ ρ<sup>n</sup> ð Þ (10)

ρð Þ i; j; k (11)

� � (13)

(12)

<sup>i</sup>¼1ΔTð Þ<sup>i</sup> (6)

<sup>Ω</sup>ð Þ<sup>i</sup> � <sup>U</sup>ð Þ <sup>0</sup> (7)

CA (8)

i¼1

The operation, translation, and rotation are done by applying these vectors:

$$\mathbf{T}^{(i+1)} = \mathbf{T}^{(i)} + \Delta \mathbf{T}^{(i+1)} = \begin{pmatrix} t\_x^{(i)} \\ t\_\mathcal{\mathcal{Y}}^{(i)} \\ t\_x^{(i)} \end{pmatrix} + \begin{pmatrix} \Delta t\_x^{(i+1)} \\ \Delta t\_\mathcal{\mathcal{Y}}^{(i+1)} \\ \Delta t\_x^{(i+1)} \end{pmatrix} \tag{4}$$

$$\mathbf{U}^{(i+1)} = \mathbf{0}^{(i+1)} \times \mathbf{U}^{(i)} = \begin{pmatrix} a\_{11}^{(i+1)} & a\_{11}^{(i+1)} & a\_{13}^{(i+1)} \\ a\_{21}^{(i+1)} & a\_{22}^{(i+1)} & a\_{23}^{(i+1)} \\ a\_{31}^{(i+1)} & a\_{32}^{(i+1)} & a\_{33}^{(i+1)} \end{pmatrix} \times \begin{pmatrix} u\_{11}^{(i)} & u\_{11}^{(i)} & u\_{13}^{(i)} \\ u\_{21}^{(i)} & u\_{22}^{(i)} & u\_{23}^{(i)} \\ u\_{31}^{(i)} & u\_{32}^{(i)} & u\_{33}^{(i)} \end{pmatrix} \tag{5}$$

and many operations can be accumulated without losing accuracy:

Figure 2.

Rigid domains as a convenient way to manipulate map objects.

Protein-Protein Docking Using Map Objects DOI: http://dx.doi.org/10.5772/intechopen.83543

2.4 Rigid domains

Molecular Docking and Molecular Dynamics

rotational matrix, U:

<sup>U</sup>ð Þ <sup>i</sup>þ<sup>1</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>U</sup>ð Þ<sup>i</sup> <sup>¼</sup>

Figure 2.

66

Because a map object contains a large amount of data, it is inconvenient to perform movement on a map itself. For example, a rotation of a map object will result in the rectangular space not parallel to the coordinate axis anymore, and new boundaries and distributions need be updated accordingly. In addition, a real system often contains more than one copy of some molecular species, and it would be very memory costing to have a map object for each copy of these species. Instead, we define a rigid domain to represent a copy of the molecular species. A rigid domain contains only the identity of the map object it represents and the position and orientation vectors related to the map object, and can be manipulated easily. A rigid domain can be understood as a mobile representation of a map object. Each rigid domain has a unique identity, and many rigid domains can represent the same map object. Figure 2 shows the map objects of the α-chain and β-chain of a TCR

Each rigid domain is defined by its map ID and its translation vector, T, and

0

B@

The operation, translation, and rotation are done by applying these vectors:

t ð Þi x t ð Þi y t ð Þi z

0

BB@

<sup>11</sup> <sup>ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup> 13

<sup>22</sup> <sup>ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup> 23

<sup>32</sup> <sup>ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup> 33

u<sup>11</sup> u<sup>12</sup> u<sup>13</sup> u<sup>21</sup> u<sup>22</sup> u<sup>23</sup> u<sup>31</sup> u<sup>32</sup> u<sup>33</sup>

1

CCA þ

1

CCA � 1

Δt ð Þ <sup>i</sup>þ<sup>1</sup> <sup>x</sup> 1

CCA

0

BB@

Δt ð Þ iþ1 y Δt ð Þ <sup>i</sup>þ<sup>1</sup> <sup>z</sup>

0

BB@

uð Þi <sup>11</sup> <sup>u</sup>ð Þ<sup>i</sup>

uð Þi <sup>21</sup> <sup>u</sup>ð Þ<sup>i</sup>

uð Þi <sup>31</sup> <sup>u</sup>ð Þ<sup>i</sup>

CA (3)

<sup>11</sup> <sup>u</sup>ð Þ<sup>i</sup> 13

1

CCA

<sup>22</sup> <sup>u</sup>ð Þ<sup>i</sup> 23

<sup>32</sup> <sup>u</sup>ð Þ<sup>i</sup> 33 (4)

(5)

variable domain and their manipulation through rigid domains.

tx ty tz 1

CA, <sup>U</sup> <sup>¼</sup>

0

B@

<sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> <sup>¼</sup> <sup>T</sup>ð Þ<sup>i</sup> <sup>þ</sup> <sup>Δ</sup>Tð Þ <sup>i</sup>þ<sup>1</sup> <sup>¼</sup>

0

BB@

Rigid domains as a convenient way to manipulate map objects.

ωð Þ <sup>i</sup>þ<sup>1</sup> <sup>11</sup> <sup>ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup>

ωð Þ <sup>i</sup>þ<sup>1</sup> <sup>21</sup> <sup>ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup>

ωð Þ <sup>i</sup>þ<sup>1</sup> <sup>31</sup> <sup>ω</sup>ð Þ <sup>i</sup>þ<sup>1</sup>

and many operations can be accumulated without losing accuracy:

T ¼

$$T^{(n)} = T^{(n-1)} + \Delta T^{(n)} = T^{(n-2)} + \Delta T^{(n)} + \Delta T^{(n-1)} = T^0) + \sum\_{i=1}^{n} \Delta T^{(i)} \tag{6}$$

$$\mathbf{U}^{(n)} = \boldsymbol{\Omega}^{(n)} \times \mathbf{U}^{(n-1)} = \boldsymbol{\Omega}^{(n)} \times \boldsymbol{\Omega}^{(n-1)} \times \mathbf{U}^{(n-2)} = \prod\_{i=1}^{n} \boldsymbol{\Omega}^{(i)} \times \mathbf{U}^{(0)} \tag{7}$$

When a map object is created, a rigid domain at origin is created for it, with

$$\mathbf{T} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \mathbf{U} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \tag{8}$$

A rigid domain can also be created for a molecular structure by linking the structure to a given map object according to the translation vector and rotation matrix from the reference coordinates to the linked structure.

$$\mathbf{X} = \mathbf{T} + \mathbf{U} \times \mathbf{X}^{(rq\dagger)} \tag{9}$$

This equation also provides a way to update the structure coordinates according to the position and orientation of a rigid domain.

### 2.5 Map comparison

Map comparison provides a target function for fitting one map into another map. Four types of cross-correlation functions [10] are provided for comparison between map objects, which are listed below.

1. Density correlation (DC)

$$DC\_{mn} = \frac{\overline{\rho\_m \rho\_n} - \overline{\rho}\_m \overline{\rho}\_n}{\delta(\rho\_m)\delta(\rho\_n)}\tag{10}$$

where

$$\overline{\rho} = \frac{\mathbf{1}}{n\_{\mathbf{x}} n\_{\mathbf{y}} n\_{\mathbf{z}}} \sum\_{i}^{n\_{\mathbf{x}}} \sum\_{j}^{n\_{\mathbf{z}}} \sum\_{k}^{n\_{\mathbf{z}}} \rho(i, j, k) \tag{11}$$

and

$$\delta(\rho) = \sqrt{\overline{\rho^2} - \overline{\rho}^2} \tag{12}$$

represent the average and fluctuation of the density distribution. DCmn is the density correlation of map m to map n. Figure 3 shows two comparison maps in two dimensions. DCmnis calculated according to map m's dimension and grid properties. The calculation runs over all grid points of map m, which are transformed and interpolated into grid points of map n to get corresponding density properties.

2. Laplacian correlation (LC)

$$LC\_{mn} = \frac{\overline{\nabla^2 \rho\_m \nabla^2 \rho\_n} - \overline{\nabla^2 \rho\_m} \overline{\nabla^2 \rho\_n}}{\delta(\nabla^2 \rho\_m) \delta(\nabla^2 \rho\_n)}\tag{13}$$

where ∇<sup>2</sup>ρ is the Laplacian filtered density derived from density distribution by the following finite difference approximation:

Figure 3. A cartoon to show the grid-threading Monte Carlo searching method.

$$\nabla^2 \rho\_{\vec{i}\vec{k}} = \rho\_{\vec{i}+1\vec{j}k} + \rho\_{\vec{i}-1\vec{j}k} + \rho\_{\vec{i}\vec{j}+1\vec{k}} + \rho\_{\vec{i}\vec{j}-1\vec{k}} + \rho\_{\vec{i}\vec{k}+1} + \rho\_{\vec{i}\vec{k}-1} - \mathsf{G}\rho\_{\vec{i}\vec{k}} \tag{14}$$

LCmn is the Laplacian correlation of map m to map n. Similar to DCmn, LCmn is calculated according to map m's dimension and grid properties.

3. Core-weighted density correlation (CWDC)

$$\text{CWDC}\_{mn} = \frac{\overline{(\rho\_m \rho\_n)\_w} - \overline{(\rho\_m)\_w (\rho\_n)\_w}}{\delta\_w (\rho\_m) \delta\_w (\rho\_n)} \tag{15}$$

CWLCmn uses Laplacian filtered density, instead of the density in the calculation. Again, only the core region of map m has contribution to the core-weighted

Energetics of molecular systems is the basis of molecular modeling. Calculation

The electric field around a molecule is described by the field map with scaled

e1φ<sup>1</sup> (20)

e1e<sup>2</sup> (22)

(21)

E ele <sup>12</sup> <sup>¼</sup> <sup>∑</sup> m<sup>1</sup>

points of object 2. The dielectric constant, ε = 80, is used for most cases.

δ2 1 δ2 2 ∑ m<sup>1</sup>

E binding

where e<sup>1</sup> is the charge at the charge map 1 and φ<sup>2</sup> is the electric field from object 2, which depends on the dielectric constant, ε, and distances from each grid

Surface interaction brings the surface together while avoiding core overlapping. The surface can be identified by low core index. We propose to use the following equation to make the surface contact favorable while overlapping unfavorable:

> C1C<sup>2</sup> 3 � �<sup>2</sup>

where C1 and C2 are the core indices of molecular 1 and 2 at each grid point and δ<sup>1</sup> and δ<sup>2</sup> are the grid intervals of map. 1 and 2, respectively. υ is the VDW interac-

> 1ffiffiffiffiffiffiffiffi δ3 1δ3 2 <sup>q</sup> <sup>∑</sup> m<sup>1</sup>

Before and after binding, the surface charge groups change from the solvation state

to the buried state and will create an energy gain we termed as desolvation energy:

Upon binding, the surface charge groups will contact with each other. The surface charge–charge interaction is different from the charge-field interaction

<sup>12</sup> <sup>¼</sup> <sup>b</sup> <sup>δ</sup><sup>2</sup>

!

� <sup>C</sup>1C<sup>2</sup> 3

of molecular interaction using map objects is the crucial step for a successful modeling or simulation study. For atomic objects interaction calculation is pairwise and is very time-consuming for large molecular assemblies. For map objects, we propose to use field interactions that can be calculated much more efficiently. We define four types of interactions to describe interaction between map objects: electric field interaction, surface charge-charge interaction, VDW interaction, and

Laplacian correlation, CWLCmn.

Protein-Protein Docking Using Map Objects DOI: http://dx.doi.org/10.5772/intechopen.83543

2.6 Molecular interactions between map objects

desolvation interaction as described below.

coordinates. The interaction with the field is

E vdw <sup>12</sup> <sup>¼</sup> <sup>4</sup><sup>υ</sup>

3. Surface charge: charge interaction

which is screened by the solvent environment:

where b is the surface interaction parameter.

4. Desolvation interaction

1. Electric field interaction

2. VDW interaction

tion parameter.

69

where ð Þ X <sup>w</sup> represents a core-weighted average of distribution property X:

$$\overline{(X)\_w} = \frac{\sum\_{i,j,k} w\_{mn}(i,j,k)X(i,j,k)}{\sum\_{i,j,k} w\_{mn}(i,j,k)} \tag{16}$$

and

$$\delta\_w(X) = \sqrt{\overline{\left(X^2\right)\_w} - \overline{\left(X\right)\_w}^2},\tag{17}$$

$$w\_{mn} = \frac{f\_m^a}{f\_m^a + k\_c f\_n^a + b} \tag{18}$$

where wmm is core-weighting function of core m to core n. Three parameters, a, b, and kc, control the dependence of the function to the core indices. We chose a = 2 and kc = 1 in this work calculations, and b is set to a very small value, say 10–6, to ensure wmn ¼ 0 when f <sup>m</sup> ¼ 0 and f <sup>n</sup> ¼ 0. Therefore, only the core region of map m has contribution to the core-weighted density correlation, CWDCmn.

4. Core-weighted Laplacian correlation (CWLC)

$$\text{CWLC}\_{mn} = \frac{\overline{\left(\nabla^2 \rho\_m \nabla^2 \rho\_n\right)\_w} - \overline{\left(\nabla^2 \rho\_m\right)\_w \left(\nabla^2 \rho\_n\right)\_w}}{\delta\_w \left(\nabla^2 \rho\_m\right) \delta\_w \left(\nabla^2 \rho\_n\right)}\tag{19}$$

### Protein-Protein Docking Using Map Objects DOI: http://dx.doi.org/10.5772/intechopen.83543

CWLCmn uses Laplacian filtered density, instead of the density in the calculation. Again, only the core region of map m has contribution to the core-weighted Laplacian correlation, CWLCmn.

### 2.6 Molecular interactions between map objects

Energetics of molecular systems is the basis of molecular modeling. Calculation of molecular interaction using map objects is the crucial step for a successful modeling or simulation study. For atomic objects interaction calculation is pairwise and is very time-consuming for large molecular assemblies. For map objects, we propose to use field interactions that can be calculated much more efficiently. We define four types of interactions to describe interaction between map objects: electric field interaction, surface charge-charge interaction, VDW interaction, and desolvation interaction as described below.

#### 1. Electric field interaction

The electric field around a molecule is described by the field map with scaled coordinates. The interaction with the field is

$$\stackrel{cl\epsilon}{E} = \sum\_{m\_1} c\_1 \rho\_1 \tag{20}$$

where e<sup>1</sup> is the charge at the charge map 1 and φ<sup>2</sup> is the electric field from object 2, which depends on the dielectric constant, ε, and distances from each grid points of object 2. The dielectric constant, ε = 80, is used for most cases.

#### 2. VDW interaction

∇2

Figure 3.

and

68

ρijk ¼ ρ<sup>i</sup>þ1jk þ ρ<sup>i</sup>�1jk þ ρijþ1<sup>k</sup> þ ρij�1<sup>k</sup> þ ρijkþ<sup>1</sup> þ ρijk�<sup>1</sup> � 6ρijk (14)

<sup>δ</sup>wð Þ <sup>ρ</sup><sup>m</sup> <sup>δ</sup><sup>w</sup> <sup>ρ</sup><sup>n</sup> ð Þ (15)

<sup>∑</sup>i,j,kwmnð Þ <sup>i</sup>; <sup>j</sup>; <sup>k</sup> (16)

, (17)

<sup>n</sup> <sup>þ</sup> <sup>b</sup> (18)

� � (19)

LCmn is the Laplacian correlation of map m to map n. Similar to DCmn, LCmn is

CWDCmn <sup>¼</sup> <sup>ρ</sup>mρ<sup>n</sup> ð Þ<sup>w</sup> � ð Þ <sup>ρ</sup><sup>m</sup> <sup>w</sup> <sup>ρ</sup><sup>n</sup> ð Þ<sup>w</sup>

where ð Þ X <sup>w</sup> represents a core-weighted average of distribution property X:

ð Þ <sup>X</sup> <sup>w</sup> <sup>¼</sup> <sup>∑</sup>i,j,kwmnð Þ <sup>i</sup>; <sup>j</sup>; <sup>k</sup> X ið Þ ; <sup>j</sup>; <sup>k</sup>

X<sup>2</sup> � �

where wmm is core-weighting function of core m to core n. Three parameters, a, b, and kc, control the dependence of the function to the core indices. We chose a = 2 and kc = 1 in this work calculations, and b is set to a very small value, say 10–6, to ensure wmn ¼ 0 when f <sup>m</sup> ¼ 0 and f <sup>n</sup> ¼ 0. Therefore, only the core region of map

wmn <sup>¼</sup> <sup>f</sup>

m has contribution to the core-weighted density correlation, CWDCmn.

ρm∇<sup>2</sup>ρ<sup>n</sup> � �

δ<sup>w</sup> ∇<sup>2</sup>ρ<sup>m</sup>

4. Core-weighted Laplacian correlation (CWLC)

CWLCmn <sup>¼</sup> <sup>∇</sup><sup>2</sup>

f a <sup>m</sup> þ kc f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a m

<sup>w</sup> � ð Þ X <sup>w</sup> <sup>2</sup> q

a

<sup>w</sup> � <sup>∇</sup><sup>2</sup>ρ<sup>m</sup> � �

� �δ<sup>w</sup> ∇<sup>2</sup>

<sup>w</sup> <sup>∇</sup><sup>2</sup>ρ<sup>n</sup> � � w

ρn

calculated according to map m's dimension and grid properties.

δwð Þ¼ X

3. Core-weighted density correlation (CWDC)

A cartoon to show the grid-threading Monte Carlo searching method.

Molecular Docking and Molecular Dynamics

Surface interaction brings the surface together while avoiding core overlapping. The surface can be identified by low core index. We propose to use the following equation to make the surface contact favorable while overlapping unfavorable:

$$\stackrel{vdw}{E}\_{12} = 4\nu \frac{\delta\_1^2}{\delta\_2^2} \sum\_{m\_1} \left( \left( \frac{\mathbf{C}\_1 \mathbf{C}\_2}{3} \right)^2 - \frac{\mathbf{C}\_1 \mathbf{C}\_2}{3} \right) \tag{21}$$

where C1 and C2 are the core indices of molecular 1 and 2 at each grid point and δ<sup>1</sup> and δ<sup>2</sup> are the grid intervals of map. 1 and 2, respectively. υ is the VDW interaction parameter.

3. Surface charge: charge interaction

Upon binding, the surface charge groups will contact with each other. The surface charge–charge interaction is different from the charge-field interaction which is screened by the solvent environment:

$$\begin{array}{c} \stackrel{binding}{E} = b \frac{\delta\_1^2}{\sqrt{\delta\_1^3 \delta\_2^3}} \sum\_{m\_1} \mathbf{e}\_1 \mathbf{e}\_2 \end{array} \tag{22}$$

where b is the surface interaction parameter.

4. Desolvation interaction

Before and after binding, the surface charge groups change from the solvation state to the buried state and will create an energy gain we termed as desolvation energy:

$$\frac{\partial \mathcal{E}^{\rm double}}{\partial \mathcal{E}}\_{12} = \kappa \delta\_1^2 \sum\_{m\_1} \left( \frac{\mathcal{C}\_1 \mathfrak{e}\_1^2}{\delta\_1^3 \left( 1 + \left( \mathcal{C}\_1 / 2 \right)^6 \right) \left( 1 + \mathcal{C}\_2^6 \right)} - \frac{\mathcal{C}\_2 \mathfrak{e}\_2^2}{\delta\_2^3 \left( 1 + \mathcal{C}\_1^6 \right) \left( 1 + \left( \mathcal{C}\_2 / 2 \right)^6 \right)} \right) \tag{23}$$

microscopy maps. We chose a T-cell receptor (TCR) variable domain (PDB code: 1a7n) as an example complex to illustrate the modeling process with map objects. The TCR variable domain is a complex of two chains, α-chain and β-chain. The two chains are first blurred into maps of the same resolution (here 15 Å) as the EM map. Then each map is fitted into the EM map to get a complex map. The complex map is projected back to atomic structures, which is the complex structure we are looking for. The root mean square (rms) deviation of the fitting result from X-ray

The structure obtained from map fitting generally is not optimized in atomic details. There are often atom overlaps or improper spacing between components. This structural mismatch can be removed by many modeling methods available in CHARMM [14, 15], such as energy minimization and simulated annealing, if the fitting result is very close to the right structure. After the minimization, the rms

The energy function is designed to have the minimum at the binding conformation. Therefore, it is possible to determine complex structures through minimizing the map interaction energy in cases where the EM complex map is not available. It should be noted that the map object assumes certain rigidity of a molecular object. Certain flexibility of loop region can be accommodated by the low-resolution characters, while large flexibilities like domain movement should be dealt with multiple map objects. Recently, this method was successfully applied in modeling of

Figure 5 shows the steps to perform an energy-based conformational search to determine complex structures. In this case, no EM map is used. The TCR chains are transferred into property maps that allow interaction between map objects to be

complex is 3 Å.

deviation is 0.97 Å.

Figure 5.

71

3.2 Complex structures from energy optimization

the peroxiredoxin (Prx) complex [16].

Protein-Protein Docking Using Map Objects DOI: http://dx.doi.org/10.5772/intechopen.83543

Derive complex structure base on map interactions.

where s is the desolvation parameter.

These interaction parameters used to define the interactions, Eqs. (20)–(23), can be derived from atomic force field or from experimental data. By fitting into energies calculated with the CHARMM force field [11], we obtained the parameters υ = 0.14 kcalÅ, b = 330 kcal/(C<sup>2</sup> Å), and s = 70 kcal/(C<sup>2</sup> Å2 ).

### 2.7 Conformational search

We implemented the grid-threading Monte Carlo searching algorithm [10] for robustly fitting rigid domains to a target map. The grid-threading Monte Carlo (GTMC) search is a combination of the grid search and Monte Carlo sampling. As shown in Figure 3, the conformational space is split into grid points, and short Monte Carlo searches are performed to identify local maximums around the grid points. The global maximum is identified among the local minimums.
