Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 17 pptx

These bonds transform the open loop linkage to a linkage with some relatively large links rigid body domains and closed kinematic loops.. The methodology presented in this paper treats t

Figure 5. Object relative position

desired finger without the need of solving any kind of inverse kinematics equations C Canudas, G Bastin, B Siciliano Given the the differential kinematics equation

˙

X
3 =
1

125X

3· L

2+3751 X

3· L

3 −2

35X

3· L

1

  ˙q1

˙

q4,



(27)

If we want to reach the point H(s1, t1), we require that the suitable velocity at the very end of the finger should be proportional to the error

at each instance V i = −0.7(X

3 − H(s1, t1)) This velocity is mapped into the phase space by means of using the Jacobian inverse Here we

use simply the pseudo-inverse with j1 = 1251 X

3 · L

2+ 3751 X

3 · L

3 and

j2 =−2

35X

3· L

1



∆q1

∆q4



= (j1∧ j2)−1 ·



V i ∧ j2

j1∧ V i



(28) Applying this control rule, one can move any of the fingers at a desired position above an object, so that an adequate grasp is accomplish

5 Results

In this section we present the experimental results of our grasping algorithm In Figure 6, the inferior images correspond to the simulated scenario and the other ones are real In this experiment the object was suspended manually above the grasping hand, simply to check whether the has been opened correctly or not We can see that for each object the algorithm manages to find the singular grasp points, so that the object is hold properly and in equilibrium Note that the found points correspond to the expected grasping points

or it is possible to implement a control law which will allow to move the

Figure 6.

6 Conclusion

Using conformal geometric algebra we show that it is possible to find three grasping points for each kind of object, based on the intrinsic information of the object The hand s kinematic and the object structure can be easily related to each other in order to manage a natural and feasible grasping where force equilibrium is always guaranteed

References

computational geometry” G Somer, editor, Geometric Computing with Clifford

Algebras, pages 27-52 Springer-Verlag Heidelberg.

the kinematics of robot manipulators Journal of Robotics Systems, 17(9):495-516.

Carlos Canudas de Wit, Georges Bastin, Bruno Siciliano (1996) Theory of Robot Control, Springer.

Arbitrary 3D Objects ICRA, pages 1890-1896 Detroit, Michigan.

Andrew T Miller, Steffen Knoop, Peter K Allen, Henrik I Christensen (2003)

”

Au-tomatic Grasp Planning Using Shape Primitives,” In Proceedings of the IEEE International Conference on Robotics and Automation, pp 1824-1829.

Grasping some objects.

Li, H Hestenes, D Rockwood A (2001).“Generalized Homogeneous coordinates for

Bayro-Corrochano, E and K¨ ahler, D (2000) Motor Algebra Approach for Computing

Ch Borst, Fischer M and Hirzinger, G (1999) A Fast and Robust Grasp Planner for,

,

, ,

.

.

Raghavendran Subramanian

Graduate Student

Dept of Mechanical Engineering

University of Connecticut

Storrs, CT – 06268

raghavendran@engr.uconn.edu

Kazem Kazerounian

Professor

Dept of Mechanical Engineering

University of Connecticut

Storrs, CT – 06268

kazem@engr.uconn.edu

Abstract In this paper, we present a new methodology to identify the rigid domains in

a protein molecule This procedure also identifies the flexible domains as well as their degree of flexibility Identification of rigid domains significantly simplifies the motion modeling procedures (such as molecular dynamics) that use geometric features of a protein as variables

Keywords:

1 Introduction

Proteins are the building blocks that play an essential role in a variety

of basic biological functions such as signal transduction, ligand binding, catalysis, regulation of activity, transport of metabolites, formation of larger assemblies and cellular locomotion Its internal motions results in conformational transitions and often relate structure to its function Hence, comprehending the protein internal motion is the key to the understanding of the structural relationship of these natural machines to their function Protein molecules have always been observed with rigid domains connected by flexible portions as shown in the figure 1 Kinematics serial chain model of proteins has been established and justified in few of our previous works (Kazerounian 2004; Kazerounian,

Kazerounian June 2002) As the long snake type serial linkage folds, new bonds are created between atoms of the residues that are not

© 2006 Springer Printed in the Netherlands.

481

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 481– 488.

OF PROTEIN MOLECULES

IN THE STUDY OF INTERNAL MOBILITY

inematics, mobilities, functions, graph theory, nano machines, closed loops

Latif et al., 2005; Kazerounian, Latif et al., 2005; Subramanian 2005;

K

neighbors These bonds transform the open loop linkage to a linkage with some relatively large links (rigid body domains) and closed kinematic loops These bonds are generally categorized as follows: 1) Hydrogen Bonds (main chain to main chain, main chain to Side chain and side chain to side chain), and 2) Disulphide bonds

To gain insight into a protein function, we must understand the are five different computational methods reported in literature to identify rigid domains of the protein Two of the methodsalso attempt to 1995) involves comparison of two conformations of a protein to identify the rigid domains in a molecule The second method (Wriggers and Schulten 1997) also compares two different conformations of the same

dynamics of a protein molecule The procedure creates an equivalent elastic network model with

atoms as masses serially

connected one after the

This mathematical treatment

yields vibrational frequency

modes of all the atoms An

atom for which all frequency

modes are computed to be

zero, will be considered as a

part of a rigid domain This method is computationally, a very expensive procedure even with a network of just CD atoms Fourth method is a variant of the third method In this method normal mode analysis Carlo simulation is used to form the trajectory of all the atoms This

requires only one conformation to identify the rigid domains in a protein molecule It uses the distance constraints between atoms due to the

a rigidity matrix which on further manipulation based on the set of rules defined under the rigidity theory, one can find rigid domains This method disregards the presence of disulphide bonds in a protein molecule which also reduces the mobility of atoms in proteins

The methodology presented in this paper treats the protein molecule

as a kinematic chain that has open as well as closed loops In the recent

kinematics and the mobility of the internal motion of the protein T here

establish the mobility of the chain F irst method (Nichols, Rose et al

protein It uses a least square technique to best fit the two con- formations Third method (Levitt, Sander et al., 1985) is based on the

other by springs

in its open and closed conformation

coupled with molecular dynamics (Doruker, Bahar et al., 2002) and Monte method too requires unreasonably excessive computation Fifth method

presence of covalent and the hydrogen bonds between them I t develops (Jacobs, Rader et al., 2001) is based on the graph theory This method

Figure 1 Ribbon view of a peptide chain

,

past, many kinematicians (Crossley 1965; Woo 1967; Manolescu 1973; type synthesis of mechanisms especially to identify non-isomorphic mechanisms and to enumerate mechanisms This method also uses graph theory based on the primary (linear) structure of the protein, and uses the atom coordinates to detect the hydrogen and disulphide bonds The resulting graph maintains the information on the connectivity of links in the protein mechanism and thereby identifies all the loops formed by hydrogen and disulphide bonds The loops that are kinematically over-constrained form rigid structures This is an iterative process that

2 Identification of the Hydrogen Bonds

Hydrogen bonds occur when two electronegative atoms interact with the same hydrogen The hydrogen atom is covalently attached to one atom (commonly called

donor), and interacts

electrostatically with the

other atom (commonly

called acceptor) This

(hydrogen) Hydrogen

bond possesses some

degree of orientational

preference and has the

characteristics of a

covalent bond (although

it is weak) Several fine

works in the literature

have focused on this

directional behavior of hydrogen bonds (Baker and Hubbard 1984; Eswar and Ramakrishnan 2000) These works have established generalized geometric characteristics for identification of the hydrogen bonds when the positional coordinates of the electronegative atoms and the hydrogen

configuration A shortfall of these data files is that the hydrogen atom positions are usually not recorded

C

O

C

H

results in the identification of all the rigid domains

Figure 2 Location of Hydrogen atom

with respect to the neighboring atoms

atoms and the proton

interaction is due to

ween the electronegative

the dipole effect

bet-atoms are known Protein Data Bank (PDB) (Berman, Westbrook et al., Mruthyunjaya and Raghavan 1979) have extensively used graph theory for

2000) offers the coordinate position of all the atoms in a protein

2.1 Hydrogen Atom Position Calculation

The chemical (directional) nature of the covalent bonds leads to a unique relative position of a hydrogen atom with respect to the positional coordinate of its neighboring atoms Hence the coordinates of a hydrogen atom can be established theoretically using coordinates of its neighbor atoms (figure 2) The detailed procedure and formulation based on figure

2 is included in Rigid body assumption in proteins has been established and justified in few of our previous works (Subramanian 2005)

2.2 Criteria to Establish Hydrogen Bonds

There are predominantly three types of hydrogen bonds observed in the protein structures They are main chain to main chain, main chain to side chain and side chain to side chain hydrogen bonds The majority of the main chain to main chain hydrogen bonds are local in nature involving less than six consecutive residues in the primary sequence of a protein As mentioned earlier, the directional nature of the hydrogen bond results in a set of geometric criteria to be established to identify the presence of hydrogen bonds These geometric criteria solely depend on the coordinate positions of two electronegative atoms and a hydrogen atom These geometric criteria are different for different sets of the electronegative atoms and the geometric conditions for identification of the hydrogen bonds are quite extensive Reference (Subramanian 2005) and the exhaustive conditions for selecting each one of the three possible hydrogen bonds, as developed by the authors

3 Identification of the Disulphide Bonds

A disulphide bridge is formed between two cysteine residues by the oxidation of their sulfur atoms to form a double bond Thus two cysteine residues connect through their sulphur atoms and form loops in the open chain In proteins disulphide bridges contribute significantly to the stability of proteins

Two parameters have been established (Sowdhamini, Srinivasan et al 1989) to identify the presence of disulphide bonds between two cysteine residues in a protein molecule They are based on the geometric features that exist between the two test residues The distance parameters include the distance between the two alpha carbon atoms and the distance between the two beta carbon atoms of two cysteine residues in the primary sequence The two geometric conditions are that the first distance mentioned lies within 3.8Å to 6.5Å and the second distance lies within 3.4Å to 4.5Å These criteria are checked for all the possible

,

combinations of any two cysteines in a protein molecule Those combinations that meet the above requirements are assumed to form the disulphide bonds

4 Application of the Graph Theory to Loop

The internal mobility of a protein chain is a function of how various links in the open chain model connect by means of hydrogen and disulphide bonds These bonds transform the open loop linkage to more complex multi closed loop system The size of the protein molecule and the large number of such bonds demands a sophisticated method of accounting for connections within the molecule Graph theory is an ideal tool for this purpose

The equivalent linkage mechanisms to protein chains can be described

as a graph with links as edges and joints as vertices and is a very useful tool to represent the connectivity between links A two dimensional matrix (commonly called as connectivity matrix) mathematically represent the connectivity between all the links Prior to the detection of the over-constrained loops from the given connectivity matrix, all the side chain links which do not participate in the loop formation will be eliminated from the connectivity matrix This will reduce the computational complexity of the problem of detecting the over-constrained closed loops

As a first step, we will eliminate all those side chain links which do not participate in the loop formation This process starts from the end link of all the side chains If the end link has only one joint, then it does not form a loop Consequently the link preceding the removed end link becomes the end link itself This procedure iteratively eliminates all the links of the side chains that do not form closed loops except the first link

of the side chains (that is connected to the two main chain links)

The graph after the previous step will have only one side chain link for all the side chains which are not involved in the loop formation As mentioned earlier, these side chains will be connected only to two main chain links and no side chain links In the second step the procedure eliminates all such first links of the side chains This requires that the side chain links be differentiated from main chain links by their index numbers This can be done by storing the index numbers of all the main chain links and all the side chain links in two different vectors This information is readily available from Protein Data Bank (PDB) files Note that in the above two stages the size and the values in the connectivity matrix changed while eliminating all the side chain links that were not part of any closed loops This will leave the graph with

Detection in a Protein Chain

none of the open ended side chain branches off the main chain Thus the complexity involved with maintaining the information of the un-influencing side chain links is avoided This also reduces the computational needs for solving the problem of detecting all the over constrained loops

The procedure we have developed to detect all the over-constrained loops involves finding all the closed loops with two links, three links, four links, five links and six links respectively, until all the over-constrained loops are detected The steps to detect loops with m links (m = 2 to 6) are

2005) The search starts

with any link and

corresponding to that

link in the connectivity

matrix, we follow the

trail of the links

connected until we arrive

back on the link we

started the search with

This indicates that the

loop is closed and a

counter keeps track of the

number of links in that

closed loop

In the repeated local

search for all the over-constrained loops, we change the connectivity matrix every time an over-constrained loop is found The changes are as follow: all the links of an over-constrained loop is replaced by a single new link, thus the rows and columns corresponding to these links are dropped from the connectivity matrix and a new row and column is appended to the connectivity matrix to represent this new link All the connections to all the links of this over-constrained loop will now be the connections to this new link

5 Results and Discussion

numerous protein molecules to identify their rigid domains and flexible portions One such numerical experiment was on the protein BPTI (PDB

briefly explained as follows The detailed algorithm for this detec-tion process is included

in reference(Subramanian

of a “1” in the row

through the detection

Figure 3 Kinematic Sketch of the protein BPTI

(1K6U) with its rigid and flexible domains

The methodology developed in this work was succesfully applied to

-ID: 1K6U) Bovine Pancreatic Trypsin Inhibitor (PDB -ID: 1K6U) is a 58 residue long protein We identified a total of 26 hydrogen bonds in the protein molecule of which 19 were main chain to main chain hydrogen bonds and the rest of the hydrogen bonds were between main chain and side chains This protein molecule was identified with 3 disulphide bonds

4 rigid domains (R1 to R4) and

5 flexible portions (F1 to F5)

The kinematic sketch for this

protein is shown in figure 3

(kinematic arrangement) and a

3-D illustration in Figure 4

The alpha helices and beta

sheets as expected formed rigid

domains or part of rigid

domains Among all the

flexible chains, 3 of them were

closed loops The degrees of

freedom for such constrained

closed loops are also reported

These are as follow: 7 for F2, 1

for F3 and 5 for F4 These

results were compared visually

with the motion of the protein molecules available in the website: The results were observed to be consistent with these motion pictures for each of the three protein molecules

(and contact) The coordinate value of all the atoms in the protein is used only to establish the location of hydrogen and disulphide bonds It also finds all the flexible portions of a protein molecule and calculates its degrees of freedom, a numerical value as a flexibility measure, for each of these flexible portions This methodology has been successfully tested on several proteins from PDB

7 References

Baker, E N and R E Hubbard (1984) Hydrogen bonding in globular proteins Prog Biophys Mol Biol 44(2): 97-179

orange and blue) represent the different

Figure 4 Color code based distinction

between the rigid domains and flexible portions of the protein, BPTI (PDB ID: 1K6U) Red portions are the flexible regions and other colors (pink, green, rigid domains in the protein

The protein molecule had

http://molmovdb.mbb.yale.edu/molmovdb/ (Echols, Milburn et al., 2003)

tify all the rigid domains in a protein identified in a PDB type format

W e have developed a computationally efficient methodology to iden-

Berman, H M., J Westbrook, et al (2000) The Protein Data Bank Nucleic Acids Res 28(1): 235-42

Crossley, F R E (1965) The permutations of Kinematic Chains of Eight Members or Less from Graph Theoretic Viewpoint Developments in Theoretical and Applied Mechanics 2: 467-487

Doruker, P., I Bahar, et al (2002) Collective deformations in proteins determined by a mode analysis of molecular dynamics trajectories POLYMER 43(2): 431-439

Echols, N., D Milburn, et al (2003) MolMovDB: analysis and visualization of conformational change and structural flexibility Nucleic Acids Research 31(1): 478-482

Eswar, N and C Ramakrishnan (2000) Deterministic features of side-chain main-chain hydrogen bonds in globular protein structures Protein Eng 13(4): 227-38

Jacobs, D J., A J Rader, et al (2001) Protein flexibility predictions using graph theory Proteins 44(2): 150-65

Kazerounian, K (2004) From mechanisms and robotics to protein conformation and drug design Journal of Mechanical Design 126(1): 40-45

Kazerounian, K (June 2002) Is Design of New Drugs a Challenge for Kinematics? Proceedings of the 8th Int Conf on Advance Robot Kinematics - ARK, Caldes de Malavalla, Spain

Kazerounian, K., K Latif, et al (2005) Protofold: A successive kinetostatic compliance method for protein conformation prediction Journal of Mechanical Design 127(4): 712-717

Kazerounian, K., K Latif, et al (2005) Nano-kinematics for analysis of protein molecules Journal of Mechanical Design 127(4): 699-711

Levitt, M., C Sander, et al (1985) Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme J Mol Biol 181(3): 423-47 Manolescu, N I (1973) A Method based on Barnov Trusses, and using Graph Theory to find the set of Planar Jointed Kinematic Chains and Mechanisms mechanism and machine theory 8(1): 3-22

Mruthyunjaya, T S and M R Raghavan (1979) Structural Analysis of Kinematic Chains and Mechanisms based on Matrix Representation ASME Journal of Mechanical Design 101: 488-494

Nichols, W L., G D Rose, et al (1995) Rigid Domains in Proteins - an Algorithmic Approach to Their Identification Proteins-Structure Function and Genetics 23(1): 38-48

Sowdhamini, R., N Srinivasan, et al (1989) Stereochemical modeling of disulfide bridges Criteria for introduction into proteins by site-directed mutagenesis Protein Eng 3(2): 95-103

Subramanian, R (2005) Calibration of Structural Variables and Mobility Analysis of Protein molecules Mechanical Engineering Department Storrs, University of Connecticut MS

Woo, L S (1967) Type Synthesis of Plane Linkages ASME Journal of Engineering for Industry: 159-172

Wriggers, W and K Schulten (1997) Protein domain movements: Detection of rigid domains and visualization of hinges in comparisons of atomic coordinates Proteins-Structure Function and Genetics 29(1): 1-14

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