Appendix8 Shape Grammars In Chapter 9 there is a reference to possible applications of shape grammars. One such application concerns artificial biological materials constructed from a finite number of specially created (stable) Holliday type moleculesb. Shape grammars might be of practical value in a number of other applications as well. A shape grammar is defined as follows. SG = (VM, VT, %, I), where VT is a finite set of terminal symbols VM is a finite set of marker symbols (that act like non-terminals) I e Vj V^ VJ is the start symbol. The production rules of the shape grammar % are phrase structured, and the rules may be applied serially, or in parallel (the same as with Lindenmeyer systems). % is composed of rules of the form u => v, where u is a mix of Vf with V^ , and v is a mix of Vj with V^. Objects in VT and VM may have any location, orientation, or scaling (think in terms of location in a Cartesian grid or some other coordinate system, rotation, and size), in two or more dimensions. As an example, VM The marker disks may be at any one of the four branches, and marks an active area. In the example that follows, these marked active areas are where ligases may act. In the following rule for /£, the four cruciform symbols on the right of the production rule are of the same size, none are rotated, and all four have the locations indicated. J. Gips, Shape Grammars and Their Uses: Artificial Perception, Shape Generation and Computer Aesthetics, 1975, Birkhftuser. b N. C. Seeman, J. Chen, S. M. Du, J. E. Mueller, Y. Zhang, T.-J. Fu, Y. Wang, H. Wang, S. Zhang, Synthetic DNA knots and catenanes, New Journal of Chemistry, 1993, 17, 739-755. 376 Emergent Computation In the following production rule in #, the symbol on the right at the top is larger, has been rotated 45°, and its location (or position) is raised (and the markers have changed, too). Ti r In the shape grammar example to be given, each cruciform symbol signifies a stable Holliday structure with overhangs (blunt ends might work too). These stable Holliday structures may be identical in terms of their base orderings, or alternatively, these cruciferous forms may vary in the ordering of their bases (as long as ligations can occur). In addition, the bases can be naturally occurring or may be artificial bases as discussed in Chapter 3. Note, even if ligases do not yet exist that will ligate artificial bases, the artificial bases need not appear within the overhangs (or within the active areas of the blunt ends). The shape language for Holliday structures is composed of planar (2- dimensional) networks of ligated Holliday structures. It is of no significance that such planar structures occur naturally as artificial structures of this sort might be found to be useful as new bioengineering materials. An actual example of a shape grammar for Holliday structures follows. Appendix 377 M, VT, *, L = { all 2-dimensional planar networks of ligated Holliday structures } Objects of interest on the left hand sides of the rules in J£ are in vT< U vTv^ u vTv£ u vTvJ, In I I "n" branches of the 4-way Holliday structure may be ligated. r4] = 1 objects: = 4 objects: r = 6 objects: JL JL = 4 objects: J#L_ jL r Jb. - Figure A.1 AH the different marked objects used for Holliday shape grammar 378 Emergent Computation Note that while a single shape of stable Holliday structure is represented here, in fact, a class of j such molecules is possible, with variable base orderings, and different base compositions, r including "artificial" bases, abasic pairs, etc. as in Chapter 3. The only requirement is that they can Note, = 1 shape be ligated. Which element from VT is selected can be associated with a probability. Figure A.2 The single object in VT for Holliday shape grammar Figure A.3 The start symbol for Holliday shape grammar 4- .A. c, where # J J JL -v c JL where ^€ VTV^ and C, is consistent Y with the left hand side Figure A.4 Rules in ^ for Holliday shape grammar Appendix 379 (continued) 4- J*L_ HJ- c J where ^e VTV^ and ^ is consistent with the left hand side Figure A.5 Rules in % for Holliday shape grammar 380 Emergent Computation (continued) ^ J where £e VjV^ and £ is consistent with the left hand side Figure A.6 Rules in ^ for Holliday shape grammar Appendix 381 J£ (continued) JL, 9 JL- J% J L —y J L J*L J L- -TL, ^ r* " nj" ~" r C #JL3 ,J*L J L C where ge VTV^ J L and £ is consistent —\jr with the left hand side Figure A.7 Rules in ^ for Holliday shape grammar 382 Emergent Computation 1R (continued) 4» JL JL nr (any symbol in VT ) J L J'L JL -•r (any symbol in VT) JL JL JL V (any symbol in VT) Figure A.8 Rules in £> for Holliday shape grammar Appendix 383 % (continued) JL (any symbol in VT) Figure A.9 Rules in 1R for Holliday shape grammar As an example, note that in the derivation steps below, the right hand side of "=>" is not consistent with the previous symbol at the left hand side of "=>". inconsistent r - inconsistent Figure A. 10 Example of right hand sides being inconsistent with the previous symbol % (continued) A number of rules, in which The last 2 rules avoid ambiguity the markers at these positions created when two ligations can may or may not be present occurr at the same point at the same time. • • • •00-- O J'LJ Figure A.ll General disambiguating rules in 1R for Holliday shape grammar 384 Emergent Computation Examples of some (not all) disambiguating rules. L L j L • J J"* » J L L_ ,JL #J J L j L_ r r~n r J*L .J i J L J • # r"n r j L r X j i_ r JL j r X j • Figure A.12 Some specific disambiguating rules in 1R for Holliday shape grammar Appendix 385 A few derivations: #1 #2 &>4> #3 Each terminal "Holliday" structure is shown in a different shade to emphasize that the particular ordering and constituent bases in these structures is distinct as each may be probabatistically selected from a class of structures in VT. This shape grammar may or may not be viewed probabalistically. example where a marker ambiguity was introduced 386 Emergent Computation disambiguate The possibly different Holliday structures in VT could include iso-C/iso- G pairs, abasic pairs, mismatched pairs, self complementing bases, permutations of base orderings, etc. If there is more than one element in VT, then any convenient probability distribution may be chosen, or a deterministic shape grammar may be constructed. An example of a parallel shape grammar derivation stepiollows. (one step) Other shape grammars may be constructed for the 2-dimensional language of all planar networks of ligated Holliday structures. It is easy to construct a far simpler shape grammar for Holliday structures: a shape grammar that is very similar to the shape grammar for replication forks (which follows). Shape grammars are appropriate in this application, as the exact base structure (bases used and their sequences) may be exactly specified for all the finite number of species in VT. Shape grammars could be a very useful tool. Appendix 387 The possibilities abound, such as ligating 2-dimensional branching structures typically observed as replication forks (provided they can be made to be sufficiently stable). Figure A.13 Beginning the branching replication fork shape grammar Figure A.14 Rules 'R for a shape grammar for replication fork planar networks 388 Emergent Computation A sample derivation: T - YT - YYT • YVYT - YYYYT • vrmr • TftYYT Appendix 389 390 Emergent Computation Recall that a shape grammar for 2-dimensional planar Holliday networks (which is much simpler than the one given can be constructed). Such a shape graamar is quite similar to the given replication fork shape grammar above. Just as with Holliday structures, different orderings of bases, including artificial bases, mismatches, abasic pairs, etc. allow for a multiplicity of such replication fork structures, and this can be made to be probabilistic. It must be emphasized that for both the case of Holliday structures as well as replication forks, empirical evidence in the laboratory of the associated planar structures is lacking. Both these cases are idealizations, intended only to show how shape grammars might be useful as a linguistic tool. Another possibility is G-quartets viewed as 3-dimensional shape grammars (G-quartets exist in extended forms at least in vitro, and are associated with telomeres, thus stability does not seem to be a problem). Another possibility are shape grammars that will generate complicated RNA shapes including hairpins, clover-leafs, bulges, etc. Such 2-dimensional shape grammars could use associated Nussinov charts to place bases at corresponding sites. Intersecting chords that connect Nussinov "flowers" could (in 3-dimensions) underlie the links and knots found in tertiary RNA structures. The last possibility to be mentioned is the application of shape grammars to biological materials based upon threose or hexose based analogues of DNA: the TNA discussed at the end of Chapter 3. However, not only would these structures have to be sufficiently stable, but in addition, ligases would then have to exist (either natural or artificial) that function with threose or hexose TNA molecules. As a closing note, graph grammars and web grammars are just beginning to be applied to DNA, RNA, and proteins. These applications in their infancy and will not be discussed.