Two models for gene assembly in ciliates

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1 Two models for gene assembly in ciliates Tero Harju 1,3, Ion Petre 2,3, and Grzegorz Rozenberg 4 1 Deartment of Mathematics, University of Turku Turku Finland harju@utu.fi 2 Deartment of Comuter Science, Åbo Akademi University Turku Finland ietre@abo.fi 3 Turku Centre for Comuter Science Turku Finland 4 Leiden Institute for Advanced Comuter Science, Leiden University Niels Bohrweg 1, 2333 CA Leiden, the Netherlands, and Deartment of Comuter Science, University of Colorado at Boulder Boulder, Co , USA rozenber@liacs.nl Abstract. Two models for gene assembly in ciliates have been roosed and investigated in the last few years. The DNA maniulations ostulated in the two models are very different: one model is intramolecular a single DNA molecule is involved here, folding on itself according to various atterns, while the other is intermolecular two DNA molecules may be involved here, hybridizing with each other. Consequently, the assembly strategies redicted by the two models are comletely different. Interestingly however, the final result of the assembly (including the assembled gene) is always the same. We comare in this aer the two models for gene assembly, formalizing both in terms of ointer reductions. We also discuss invariants and universality results for both models. 1 Introduction Ciliates are unicellular eukaryotic organisms, see, e.g. [23]. This is an ancient grou of organisms, estimated to have originated around two billion years ago. It is also a very diverse grou some 8000 secies are currently known and many others are likely to exist. Their diversity can be areciated by comaring their genomic sequences: some ciliate tyes differ genetically more than humans differ from fruit flies! Two characteristics unify ciliates as a single grou: the ossession of hairlike cilia used for motility and food cature, and the resence of two kinds of functionally different nuclei in the same cell, a micronucleus and a macronucleus, see [15], [24], [25]; the latter feature is unique to ciliates. The macronucleus is the household nucleus all RNA transcrits are roduced in the macronucleus. The micronucleus is a germline nucleus and has no known function in the growth or in the division of the cell. The micronucleus is activated only in the rocess of sexual reroduction, where at some stage the micronuclear 1

2 genome gets transformed into the macronuclear genome, while the old macronuclear genome is destroyed. This rocess is called gene assembly, it is the most involved DNA rocessing known in living organisms, and it is most sectacular in the Stichotrichs secies of ciliates (which we consider in this aer). What makes this rocess so comlex is the unusual rearrangements that ciliates have engineered in the structure of their micronuclear genome. While genes in the macronucleus are contiguous sequences of DNA laced (mostly) on their own molecules (and some of them are the shortest DNA molecules known in Nature), the genes in the micronucleus are laced on long chromosomes and they are broken into ieces called MDSs, searated by noncoding blocks called IESs, see [15, 21 26]. Adding to the comlexity, the order of the MDSs is ermuted and MDSs may be inverted. One of the amazing features of this rocess is that ciliates aear to use linked lists in gene assembly, see [29, 30], similarly as in software engineering! Two different models have been roosed for gene assembly. The first one, roosed by Landweber and Kari, see [19], [20], is intermolecular: the DNA maniulations here may involve two molecules exchanging arts of their sequences through recombination. The other one, roosed by Ehrenfeucht, Prescott, and Rozenberg, see [11], [27], is intramolecular: here, all maniulations involve one single DNA molecule folding on itself and swaing arts of its sequence through recombination. In the intermolecular model one traditionally attemts to cature both the rocess of identifying ointers and the rocess of using ointers by oerations that accomlish gene assembly. In the intramolecular model one assumes that the ointer structure of a molecule is known, i.e., the ointers have been already identified. This imlies some imortant differences between the models: e.g., the intramolecular reresentations of genes contain only ointers, with two occurrences for each ointer, and moreover, rocessing a ointer imlies its removal from the rocessed string; these roerties do not hold in the intermolecular model. Finally, the bulk of the work on the intermolecular model, see [1 3, 16 18], is concerned with the comutational ower of the oerations in the sense of comutability theory; e.g., it is roved in [18 20] that the model has the comutational ower of the Turing machine. On the other hand, research on the intramolecular model, see [4 6, 8 10, 12 14] and esecially [7], deals with reresentations and roerties of the gene assembly rocess (reresented by various kinds of reduction systems). We believe that the two aroaches together shed light on the comutational nature of gene assembly in ciliates. In this aer, we take a novel aroach on the intermolecular model aiming to comare the assembly strategies redicted by each model. Therefore, we formalize both models in terms of MDS-IES descritors and describe the gene assembly in terms of ointer reductions. We rove a universality result showing that the assembly ower of the two models is the same: any gene that can be assembled in one model can also be assembled in the other. Nevertheless, the assembly strategies and the gene atterns throughout the rocess are comletely different in the two models. Somewhat surrisingly, we show that the two models agree on the final results of the assembly rocess. 2

3 2 The structure of micronuclear genes We shall now take a formal aroach to gene assembly. The central role in this rocess is layed by ointers. These are short sequences at the ends of MDSs (i.e., at the border of an MDS and an IES) the ointer in the end of an MDS M coincides as a nucleotide sequence with the ointer in the beginning of the MDS following M in the macronuclear gene, see [22, 25]. For the urose of an adequate formal reresentation, the first (last, res.) MDS begins (ends, res.) with a secific marker b (e, res.). It is enough for our uroses to describe any MDS by the air of ointers or markers flanking it at its ends. The gene will then be described as a sequence of such airs intersersed with strings describing the sequence of IES we thus obtain MDS-IES descritors formally defined in the following. For more details we refer to [7]. For an alhabet Σ and a string u over Σ, we will denote by [u] the circular version of string u we refer to [7] for a formal definition. Let Σ = {a a Σ} and u = a 1 a 2... a n, a i Σ Σ. The inverse of u is the string u = a n... a 2 a 1, where a = a, for all a Σ. The emty string will be denoted by Λ. Let M = { b, e, b, e } denote the set of the markers and their inverses. For each index κ 2, let κ = { 2, 3,..., κ } and Π κ = κ κ. An element Π κ is called a ointer. Also let Γ κ = { (b, e), } { (b, i), (i, e) 2 i κ } { (i, j) 2 i < j κ } and Γ κ = {(β, α) (α, β) Γ κ }. A string δ over Γ κ Γ κ is called an MDS descritor if (a) δ has exactly one occurrence from the set {b, b} and exactly one occurrence from the set {e, e}; (b) δ has either zero, or two occurrences from {, }, for any ointer Π κ. Let Ω κ = {I 1, I 2,..., I κ 1 } and Ω κ = {I I Ω κ }. Any string ι over Ω κ Ω κ is called an IES-descritor if for any I Ω κ, ι contains at most one occurrence from {I, I}. A string γ over Γ κ Γ κ Ω κ Ω κ is called an MDS-IES descritor if γ = ι 1 ( 1, q 1 )ι 2 ( 2, q 2 )... ι n ( n, q n )ι n+1, where ι 1 ι 2... ι n+1 is an IES-descritor, and ( 1, q 1 )... ( n, q n ) is an MDS-descritor. We say that γ is assembled if γ = ι 1 (m, m )ι 2 for some IES-descritors ι 1, ι 2 and m, m M. If (m, m ) = (b, e), then we say that γ is assembled in the orthodox order and if (m, m ) = (e, b), then we say that γ is assembled in the inverted order. A circular string [γ] is an (assembled) MDS-IES descritor if γ is so. Examle 1. The MDS-IES descritor associated to the micronuclear actin I gene in S.nova, shown in Fig. 1, is M 3 I 1 M 4 I 2 M 6 I 3 M 5 I 4 M 7 I 5 M 9 I 6 M 2 I 7 M 1 I 8 M 8. 3

4 Fig. 1. Structure of the micronuclear gene encoding actin rotein in the stichotrich Sterkiella nova. The nine MDSs are in a scrambled disorder. 3 Two models for gene assembly We briefly resent in this section the intramolecular and the intermolecular models for gene assembly in ciliates. We then formalize both models in terms of ointer reductions and MDS-IES descritors. For more details we refer to [7, 11, 19, 20, 27]. 3.1 The intramolecular model Three intramolecular oerations were ostulated in [11] and [27] for gene assembly: ld, hi, and dlad. In each of these oerations, a linear DNA molecule containing a secific attern is folded on itself in such a way that recombination can take lace. Oerations hi and dlad yield as a result a linear DNA molecule, while ld yields one linear and one circular DNA molecule, see Figs The secific atterns required by each oeration are described bellow: (a) (b) (c) (d) Fig. 2. Illustration of the ld molecular oeration. (a) (b) (c) (d) Fig. 3. Illustration of the hi molecular oeration. (i) The ld oeration is alicable to molecules in which two occurrences (on the same strand) of the same ointer flank one IES. The molecule is folded so that the two occurrences of are aligned to guide the recombination, see Fig. 2. As a result, one circular molecule is excised. 4

5 (a) (b) (c) Fig. 4. Illustration of the dlad molecular oeration. (ii) The hi oeration is alicable to molecules in which a ointer has two occurrences, of which exactly one is inverted. The folding is done as in Fig. 3 so that the two occurrences of are aligned to guide the recombination. (iii) The dlad oeration is alicable to molecules in which two ointers and q have intersersed occurrences (on the same strand): q q. The folding is done as in Fig. 4 so that the two occurrences of and q are aligned to guide the double recombination. Oerations ld, hi, and dlad can be formalized in terms of reduction rules ld, hi, and dlad for MDS-IES descritors as follows: (1) For each Π κ, the ld-rule for is defined by: ld (δ 1 (q, )ι 1 (, r)δ 2 ) = δ 1 (q, r)δ 2 + [ι 1 ], ld (ι 1 (, m)ι 2 (m, )ι 3 ) = ι 1 ι 3 + [(m, m)ι 2 ], where q, r Π κ M, δ 1, δ 2 are MDS-IES descritors, ι 1, ι 2, ι 3 are IES descritors, and m, m M. (2) For each Π κ, the hi-rule for is defined by: hi (δ 1 (, q)δ 2 (, r)δ 3 ) = δ 1 δ 2 (q, r)δ 3, hi (δ 1 (q, )δ 2 (r, )δ 3 ) = δ 1 (q, r)δ 2 δ 3, where q, r Π κ and δ 1, δ 2 (Γ κ Ω). (3) For each, q Π κ, q, the dlad rule for and q is defined by: dlad,q (δ 1 (, r 1 )δ 2 (q, r 2 )δ 3 (r 3, )δ 4 (r 4, q)δ 5 ) = δ 1 δ 4 (r 4, r 2 )δ 3 (r 3, r 1 )δ 2 δ 5, dlad,q (δ 1 (, r 1 )δ 2 (r 2, q)δ 3 (r 3, )δ 4 (q, r 4 )δ 5 ) = δ 1 δ 4 δ 3 (r 3, r 1 )δ 2 (r 2, r 4 )δ 5, dlad,q (δ 1 (r 1, )δ 2 (q, r 2 )δ 3 (, r 3 )δ 4 (r 4, q)δ 5 ) = δ 1 (r 1, r 3 )δ 4 (r 4, r 2 )δ 3 δ 2 δ 5, dlad,q (δ 1 (r 1, )δ 2 (r 2, q)δ 3 (, r 3 )δ 4 (q, r 4 )δ 5 ) = δ 1 (r 1, r 3 )δ 4 δ 3 δ 2 (r 2, r 4 )δ 5, dlad,q (δ 1 (, r 1 )δ 2 (q, )δ 4 (r 4, q)δ 5 ) = δ 1 δ 4 (r 4, r 1 )δ 2 δ 5, dlad,q (δ 1 (, q)δ 3 (r 3, )δ 4 (q, r 4 )δ 5 ) = δ 1 δ 4 δ 3 (r 3, r 4 )δ 5, dlad,q (δ 1 (r 1, )δ 2 (q, r 2 )δ 3 (, q)δ 5 ) = δ 1 (r 1, r 2 )δ 3 δ 2 δ 5, where r 1, r 2, r 3, r 4, r 5 Π κ, and δ 1, δ 2, δ 3, δ 4, δ 5 (Γ κ Ω). Note that each oeration removes one or two ointers from the MDS-IES descritor. When assembled (on a linear or on a circular string), the descritor 5

6 has no ointers anymore. Thus, the whole rocess of gene assembly may be viewed as a rocess of ointer removals. If a comosition ϕ of ld, hi, and dlad oerations is alicable to an MDS- IES descritor γ, then ϕ(γ) is a set of linear and circular MDS-IES descritors. We say that ϕ is a successful reduction for γ if no ointers occur in any of the descritors in ϕ(γ). Examle 2. Consider the MDS-IES descritor δ = (b, 2)I 1 (2, 3)I 2 (4, e)i 3 (3, 4). An assembly strategy for this descritor in the intramolecular model is the following: dlad 3,4 (δ) = (b, 2)I 1 (2, e)i 3 I 2, ld 2 (dlad 3,4 (δ)) = (b, e)i 3 I 2 + [I 1 ]. 3.2 The intermolecular model Three oerations were ostulated in [19] and [20] for gene assembly. One of these oerations is intramolecular: it is a sort of a generalized version of the ld oeration, while the other two are intermolecular: they involve recombination between two different DNA molecules, linear or circular, see Figs We describe these oerations below in terms of ointers, similarly as for the intramolecular model. (i) In the first oeration a DNA molecule containing two occurrences of the same ointer x (on the same strand) is folded so that they get aligned to guide the recombination, see Fig. 5. Note that unlike in ld, the two occurrences of x may have more than just one IES between them. (ii) The second oeration is the inverse of the first one: two DNA molecules, one linear and one circular, each containing one occurrence of a ointer x get aligned so that the two occurrences of x guide the recombination, yielding one linear molecule see Fig. 5. (iii) The third oeration is somewhat similar to the second one: two linear DNA molecules, each containing one occurrence of a ointer x get aligned so that the two occurrences of x guide the recombination, yielding two linear molecules, see Fig. 6. Note that the three molecular oerations in this model are reversible, unlike the oerations in the intramolecular model this is one of the main differences between the two models. We formalize now this intermolecular model in terms of reduction rules for MDS-IES descritors. The three oerations defined above are modelled by the following reduction rules for MDS-IES descritors: 6

7 x x Fig. 5. Illustration of the intramolecular oeration of the Landweber-Kari model. x Fig. x 6. Illustration of the intermolecular oeration of the Landweber-Kari model. δ 1 (q, )δ 2 (, r)δ 3 δ 1 (, q)δ 2 (r, )δ 3 δ 1 (, q)δ 2 + [(r, )δ 3 ] δ 1 (q, )δ 2 + [(, r)δ 3 ] δ 1 (, q)δ 2 + δ 3 (r, )δ 4 δ 1 (q, r)δ 3 + [δ 2 ], (1) δ 1 δ 3 + [δ 2 (r, q)], (2) δ 1 δ 3 (r, q)δ 2, (3) δ 1 (q, r)δ 3 δ 2, (4) δ 1 δ 4 + δ 3 (r, q)δ 2, (5) where δ 1, δ 2, δ 3 (Γ κ Γ κ Ω κ Ω κ ). Note that each reduction rule above removes one ointer, thus making the whole rocess irreversible. Although the intermolecular model was secifically intended to be reversible, this restriction hels in unifying the notation for (and the reasoning about) the two models and it suffices for the results resented in this aer. If a comosition ϕ of the reduction rules (1)-(5) is alicable to an MDS- IES descritor γ, then ϕ(γ) is a set of linear and circular MDS-IES descritors. We say that ϕ is a successful reduction for γ if no ointers occur in any of the descritors in ϕ(γ). Examle 3. Consider the MDS-IES descritor δ = (b, 2)I 1 (2, 3)I 2 (4, e)i 3 (3, 4) of Examle 2. An assembly strategy for this descritor in the intermolecular model is the following: δ (b, 2)I 1 (2, 4) + [I 2 (4, e)i 3 ] (b, 2)I 1 (2, e)i 3 I 2 (b, e)i 3 I 2 + [I 1 ]. Note that although the assembly strategy is very different from the one in Examle 2, the final result of the assembly, {(b, e)i 3 I 2, [I 1 ]} is the same in the two models. 7

8 4 Reduction strategies in the two models The obvious difficulty with the intermolecular model is that it cannot deal with DNA molecules in which a ointer is inverted this is the case, e.g., for the actin I gene in S.nova. Nevertheless, we can show that inverted ointers can be handled in this model, rovided the inut molecule (or its MDS-IES descritor) is available in two coies. Moreover, we consider all linear descritors modulo inversion. The first assumtion is essentially used in research on the intermolecular model, see [16 18, 20]. The second assumtion is quite natural whenever we model double-stranded DNA molecules. As a matter of fact, we use the two assumtions to conclude that for each inut descritor, both the descritor and its inversion are available. Then the hi-rule can be simulated using the intermolecular rules as follows. Let δ = δ 1 (, q)δ 2 (, r)δ 3 (the other case is treated similarly) be an MDS- IES descritor to which hi is alicable. Therefore, we assume that also δ = δ 3 (r, )δ 2 (q, )δ 1 is available. Then we obtain δ + δ δ 1 δ 2 (q, ) δ 1 + δ 3 (r, q)δ 2 (, r) δ 3 δ 1 δ 2 (q, r)δ 3 + δ 3 (r, q)δ 2 δ 1 = hi (δ) + hi (δ). Note that, having two coies of the initial string available, this rule yields two coies of hi (w). We also observe that the ld-rule is a articular case of intermolecular rules (1) and (2), obtained by setting δ 2 = Λ. Moreover, the dlad-rule can be simulated using intermolecular rules as follows. Let δ = δ 1 (, r 1 )δ 2 (q, r 2 )δ 3 (r 3, )δ 4 (r 4, q)δ 5 ) be an MDS-IES descritor to which dlad,q is alicable all other cases can be treated similarly. Then δ δ 1 δ 4 (r 4, q)δ 5 + [δ 2 (q, r 2 )δ 3 (r 3, r 1 )] = δ 1 δ 4 (r 4, q)δ 5 + [δ 3 (r 3, r 1 )δ 2 (q, r 2 )] q δ 1 δ 4 (r 4, r 2 )δ 3 (r 3, r 1 )δ 2 δ 5 = dlad,q (w). The following results is thus roved. Theorem 1. Let δ be an MDS-IES descritor having a successful reduction in the intramolecular model. If two coies of δ are available, then δ has a successful reduction in the intermolecular model. The following universality result has been roved in [8], see [7] for more details. Theorem 2. Any MDS-IES descritor has a successful reduction in the intramolecular model. Theorems 1 and 2 give the following universality result for the intermolecular model. Corollary 1. Any MDS-IES descritor available in two coies has a successful reduction in the intermolecular model. 8

9 5 Invariants In the following two examles we consider the actin I gene in S.nova and investigate assembly strategies for this gene in the intra- and inter-molecular models. Examle 4. Consider the actin gene in Sterkiella nova, see Fig. 1, having the MDS IES descritor δ = (3, 4)I 1 (4, 5)I 2 (6, 7)I 3 (5, 6)I 4 (7, 8)I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 (8, 9). Consider then an assembly strategy for δ, e.g., ld 4 dlad 5,6 ld 7 dlad 8,9 hi 2 hi 3 : ld 4 (δ) = (3, 5)I 2 (6, 7)I 3 (5, 6)I 4 (7, 8)I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 (8, 9) + [I 1 ], dlad 5,6 (ld 4 (δ)) = I 0 (3, 7)I 3 I 2 I 4 (7, 8)I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 (8, 9) + [I 1 ], ld 7 (dlad 5,6 (ld 4 (δ))) = (3, 8)I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 (8, 9) + [I 1 ] + [I 3 I 2 I 4 ], dlad 8,9 (ld 7 (dlad 5,6 (ld 4 (δ)))) = (3, e)i 6 (3, 2)I 7 (b, 2)I 8 I 5 + [I 1 ] + [I 3 I 2 I 4 ], hi 2 (dlad 8,9 (ld 7 (dlad 5,6 (ld 4 (δ))))) = (3, e)i 6 (3, b)i 7 I 8 I 5 + [I 1 ] + [I 3 I 2 I 4 ], hi 3 (hi 2 (dlad 8,9 (ld 7 (dlad 5,6 (ld 4 (δ)))))) = I 6 (e, b)i 7 I 8 I 5 + [I 1 ] + [I 3 I 2 I 4 ]. Thus, the gene is assembled in the inverted order, laced in a linear DNA molecule, with the IES I 6 receding it and the sequence of IESs I 7 I 8 I 5 succeeding it. Two circular molecules are also roduced: [I 1 ] and [I 3 I 2 I 4 ]. Examle 5. Consider the same actin gene in Sterkiella nova with the MDS IES descritor δ = (3, 4)I 1 (4, 5)I 2 (6, 7)I 3 (5, 6)I 4 (7, 8)I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 (8, 9). Then δ can be assembled in the intermolecular model as follows: δ 5 (3, 4)I 1 (4, 6)I 4 (7, 8)I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 (8, 9) + [I 2 (6, 7)I 3 ] 8 (3, 4)I 1 (4, 6)I 4 (7, 9) + [I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 ] + [I 2 (6, 7)I 3 ] 4 (3, 6)I 4 (7, 9) + [I 1 ] + [I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 ] + [I 2 (6, 7)I 3 ] 7 (3, 6)I 4 I 3 I 2 (6, 9) + [I 1 ] + [I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 ] 6 (3, 9) + [I 4 I 3 I 2 ] + [I 1 ] + [I 5 (9, e)i 6 (3, 2)I 7 (b, 2)I 8 ] 9 (3, e)i 6 (3, 2)I 7 (b, 2)I 8 I 5 + [I 4 I 3 I 2 ] + [I 1 ]. Since δ is also available, the assembly continues as follows. Here, for a (circular) string τ, we use 2 τ to denote τ + τ: 9

10 δ + δ... (3, e)i 6 (3, 2)I 7 (b, 2)I 8 I 5 + I 5 I 8 (2, b)i 7 (2, 3)I 6 (e, 3) + 2 [I 4 I 3 I 2 ] + 2 [I 1 ] 2 (3, e)i 6 (3, b)i 7 (2, 3)I 6 (e, 3) + I 5 I 8 I 7 (b, 2)I 8 I [I 4 I 3 I 2 ] + 2 [I 1 ] 2 (3, e)i 6 (3, b)i 7 I 8 I 5 + I 5 I 8 I 7 (b, 3)I 6 (e, 3) + 2 [I 4 I 3 I 2 ] + 2 [I 1 ] 3 (3, e)i 6 + I 5 I 8 I 7 (b, 3)I 6 (e, b)i 7 I 8 I [I 4 I 3 I 2 ] + 2 [I 1 ] 3 I 6 (e, b)i 7 I 8 I 5 + I 5 I 8 I 7 (b, e) + 2 [I 4 I 3 I 2 ] + 2 [I 1 ] = 2 (I 6 (e, b)i 7 I 8 I 5 + +[I 1 ] + [I 3 I 2 I 4 ]). Note that this intermolecular assembly redicts the same context for the assembled string, the same set of residual strings, and the same linearity of the assembled string as the intramolecular assembly considered in Examle 4. It is clear from the above two examles, see also Examles 2 and 3, that the two models for gene assembly redict comletely different assembly strategies for the same micronuclear gene. As it turns out however, the redicted final result of the assembly, i.e., the linearity of the assembled gene and the exact nucleotide sequences of all excised molecules, is the same in the two models, see [7] for details. The following is a result from [7], see also [10]. Theorem 3. Let δ be an MDS IES descritor. If ϕ 1 and ϕ 2 are any two successful assembly strategies for δ, intra- or inter-molecular, then (1) if ϕ 1 (δ) is assembled in a linear descritor, then so is ϕ 2 (δ); (2) if ϕ 1 (δ) is assembled in a linear descritor in orthodox order, then so is ϕ 2 (δ); (3) The sequence of IESs flanking the assembled gene is the same in ϕ 1 (δ) and ϕ 2 (δ); (4) The sequence of IESs in all excised descritors is the same in ϕ 1 (δ) and ϕ 2 (δ); (5) There si an equal number of circular descritors in ϕ 1 (δ) and ϕ 2 (δ). Examle 6. Consider the MDS IES descritor δ = (10, 8)I 1 (3, b)i 2 (5, 3)I 3 (10, 11)I 4 (5, 8)I 5 (11, e). A successful assembly strategy for δ in the intramolecular model is the following: hi 10 (δ) = I 3 (3, 5)I 2 (b, 3)I 1 (8, 11)I 4 (5, 8)I 5 (11, e), dlad 8,11 (hi 10 (δ)) = I 3 (3, 5)I 2 (b, 3)I 1 I 5 I 4 (5, e), dlad 3,5 (dlad 8,11 (hi 10 (δ))) = I 3 I 1 I 5 I 4 I 2 (b, e). 10

11 Thus, δ is always assembled in a linear molecule, and no IES is excised during the assembly rocess, i.e., no ld is ever alied in a rocess of assembling δ. Moreover, the assembled descritor will always be receded by the IES sequence I 3 I 1 I 5 I 4 I 2 and followed by the emty IES sequence. Examle 7. Consider the MDS IES descritor δ = (10, 8)I 1 (3, b)i 2 (5, 3)I 3 (10, 11)I 4 (5, 8)I 5 (11, e) from Examle 6. Then δ can be assembled in the intermolecular model as follows: δ 3 (10, 8)I 1 I 3 (10, 11)I 4 (5, 8)I 5 (11, e) + [I 2 (5, b)] 11 (10, 8)I 1 I 3 (10, e) + [I 4 (5, 8)I 5 ] + [I 2 (5, b)]. Since also δ is available, the assembly continues as follows: δ + δ... (10, 8)I 1 I 3 (10, e) + (e, 10)I 3 I 1 (8, 10) [I 4 (5, 8)I 5 ] + 2 [I 2 (5, b)] I 3 I 1 (8, 10) + (e, 8)I 1 I 3 (10, e) + 2 [I 4 (5, 8)I 5 ] + 2 [I 2 (5, b)] I 3 I 1 (8, e) + (e, 8)I 1 I 3 + [I 4 (5, 8)I 5 ] + [I 5 (8, 5)I 4 ] + 2 [I 2 (5, b)] 8, 8 I 3 I 1 I 5 I 4 (5, e) + (e, 5)I 4 I 5 I 1 I 3 + [(5, b)i 2 ] + [I 2 (b, 5)] 5, 5 I 3 I 1 I 5 I 4 I 2 (b, e) + (e, b)i 2 I 4 I 5 I 1 I 3 = 2 I 3 I 1 I 5 I 4 I 2 (b, e). Note that, again, we obtain the same context for the assembled string, the same set of residual strings, and the same linearity of the assembled string as the intramolecular assembly considered in Examle 6. References 1. Daley, M., Comutational Modeling of Genetic Processes in Stichotrichous Ciliates. PhD thesis, University of London, Ontario, Canada (2003) 2. Daley, M., and Kari, L., Some roerties of ciliate bio-oerations. Lecture Notes in Comut. Sci (2003) Daley, M., Ibarra, O. H., and Kari, L., Closure roeties and decision questions of some language classes under ciliate bio-oerations. Theoret. Comut. Sci., to aear 4. Ehrenfeucht, A., Harju, T., Petre, I., Prescott, D. M., and Rozenberg, G., Formal systems for gene assembly in ciliates. Theoret. Comut. Sci. 292 (2003) Ehrenfeucht, A., Harju, T., Petre, I., and Rozenberg, G., Patterns of micronuclear genes in cliates. Lecture Notes in Comut. Sci (2002)

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13 24. Prescott, D. M., The evolutionary scrambling and develomental unscabling of germlike genes in hyotrichous ciliates. Nucl. Acids Res. 27 (1999), Prescott, D. M., Genome gymnastics: unique modes of DNA evolution and rocessing in ciliates. Nat. Rev. Genet. 1(3) (2000) Prescott, D. M., and DuBois, M., Internal eliminated segments (IESs) of Oxytrichidae. J. Eukariot. Microbiol. 43 (1996) Prescott, D. M., Ehrenfeucht, A., and Rozenberg, G., Molecular oerations for DNA rocessing in hyotrichous ciliates. Euro. J. Protistology 37 (2001) Prescott, D. M., Ehrenfeucht, A., and Rozenberg, G., Temlate-guided recombination for IES elimination and unscrambling of genes in stichotrichous ciliates. Technical Reort , LIACS, Leiden University (2002) 29. Prescott, D. M., and Rozenberg, G., How ciliates maniulate their own DNA A slendid examle of natural comuting. Natural Comuting 1 (2002) Prescott, D. M., and Rozenberg, G., Encryted genes and their reassembly in ciliates. In: M. Amos (ed.) Cellular Comuting, Oxford University Press, Oxford (2003) 13