Theories and models in symbiogenesis - CiteSeerX

Nonlinear Analysis: Real World Applications 4 (2003) 743 – 753 www.elsevier.com/locate/na

Theories and models in symbiogenesis P.L. Antonellia;∗ , L. Bevilacquab , S.F. Rutzc a Department

of Mathematical Sciences, University of Alberta, Edmonton, Alta., T6G 2G1, Canada Nacional de Computac%a˜ o Cient"'(ca, CxP 95113, Petr"opolis, RJ 25651-070, Brazil c Department of Mathematical Sciences, University of Alberta, Edmonton, Alta., T6G 2G1, Canada b Laborat" orio

Received 9 August 2002; received in revised form 23 August 2002

Abstract Two modern theories of biological evolution, one by Carl Woese, and the other by Lynn Margulis, are modelled with Volterra–Hamilton systems. Their predictions are evaluated and compared within this modelling framework. For example, Woese’s theory turns out to su8er from instability in its chemical exchanges processes, whereas Margulis’ does not. An introduction to the mathematical and biological ideas is included. ? 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction Symbiosis is an association between two or several speci:cally distinct organisms. This association, completed by coadaptation and coevolution, leads to the formation of a new biological entity called a symbiocosm [15]. From a genetical point of view, symbiosis is a mechanism to acquire new genes by lateral transfer, i.e., exchanging bits of RNA and DNA. In successful symbioses, evolution is towards more integrated metabolism for the whole. The integration of an endosymbiont into its host metabolism is often so high that the symbiont appears much like a cytoplasmic organelle [14]. The mathematical results presented herein are an argument in favour of the Serial Endosymbiosis Theory [13] according to which the mitochondria and plastids are of symbiotic origin, somatic cells of modern plants and animals having evolved from separately living bacterial species. The formation of new species is produced at di8erent moments in the history of life, and it is from there that they evolve, and at such times evolution usually moves towards ∗

Corresponding author. Tel.: +1-403-492-5731; fax: +1-403-492-6826. E-mail addresses: [email protected] (P.L. Antonelli), [email protected] (L. Bevilacqua), [email protected] (S.F. Rutz). 1468-1218/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1468-1218(02)00085-8

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P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 4 (2003) 743 – 753

increasing integration of the separate partners. According to Nardon and Grenier [15], there are :ve types of association which we may term pre-symbiont partnerships. These are: (a) (b) (c) (d)

parasitism, which can evolve towards a neutral association; primary symbioses, where the partners cohabit in a more or less neutral fashion; secondary symbioses, which are symbioses occuring in the digestive tract; perfectly integrated intracellular symbioses, where the symbiont is perfectly localized in the host and its density strictly host-controlled; (e) organo-genetic symbioses, where the symbiont, an ancient bacterium, is transformed into a cellular organelle. Here, we will be concerned primarily with (d) and (e) above. In these cases, the pre-symbiont partnership entails the average proliferation rate of cells in each of the two partner populations being equal to allow, by means of their respective RNA, DNA replication machinery, an association, in which all members of the smaller type (e.g., mitochondria) can live inside the larger in a one-to-one fashion. We shall be modelling their chemical exchange (RNA, DNA) mathematically as information exchange. The evolutionary theory by Carl Woese [6], which is meant to address the recently discovered plethora of novel genes in ancestral genomes, unlike Lynn Margulis’, is not based on a symbiotic evolutionary mechanism. At the present time, thirty or so genomes, containing far more novel genes than had been expected, have been decoded. If all these genes had forebears in the :rst ancestor, the primeval cell would have been implausibly complex. Therefore some genes must have been transfered laterally between lineages of organisms. Presumably, this is much the same as modern bacteria exchanging genes to confer resistance to antibiotics. Bits of information would be exchanged via transfer of short modules (small pieces of RNA) carrying several related genes. Woese proposed the Ancestral Commune Theory to explain the complexity of the universal ancestor. This primeval ancestor would have been, not a single cell, but a loosely knit conglomerate of bacterial species, exchanging genetic information, module by module. We shall similarly model Woese’s theory and contrast it with our model of Margulis’ theory. In the next section, we give a brief description of the geometric theory of second order systems of di8erential equations, known as KCC-theory, upon which we shall base our treatment of Volterra-Hamilton theory. Afterwards, we will proceed to the modelling and comparisons. 2. Geometrical background Let (x1 ; : : : ; xn )=(x), (d x1 =dt; : : : ; d xn =dt)=(d x=dt)=(x), ˙ and t be 2n+1 coordinates in an open connected subset of the Euclidean (2n+1)-dimensional space Rn ×Rn ×R1 . Suppose that we have d 2 xi + gi (x; x; ˙ t) = 0; dt 2

i = 1; : : : ; n:

(1)


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for which each gi is C ∞ in a neighbourhood of initial conditions ((x)0 ; (x) ˙ 0 ; t0 ) ∈ . The intrinsic geometric properties of (1) under transformations of the type i xJ = fi (x1 ; : : : ; xn ); i = 1; : : : ; n; (2) tJ = t; are given by the :ve KCC-di8erential invariants, named after Kosambi [10], Cartan [4], and Chern [5], given below. Let us :rst de:ne the KCC-covariant di>erential of a contravariant vector :eld i (x) on by d i 1 D i = + g;i r r ; 2 dt dt

(3)

where the semi-colon indicates partial di8erentiation with respect to x˙r , and we made use of the Einstein summation convention on repeated indices. Using (3), Eq. (1) becomes 1 Dx˙i = i = g;i r x˙r − gi ; 2 dt

(4)

de:ning the (rst KCC-invariant of (1), the contravariant vector :eld on , i , which represents an ‘external force’. Varying trajectories xi (t) of (1) into nearby ones according to xJi (t) = xi (t) + i (t);

(5)

where denotes a constant with || small and i (t) are the components of some contravariant vector de:ned along xi = xi (t), we get, substituting (5) into (1) and taking the limit as → 0, i d 2 i i d + g + g;i r r = 0; ;r dt 2 dt

(6)

where the comma indicates partial di8erentiation with respect to xr . Using the KCCcovariant di8erentiation (3) we can re-express this as D 2 i = Pri r ; dt 2

(7)

where Pji = −g;i j −

1 r i 1 1 1 @ i g g; r ; j + x˙r g;i r; j + g;i r g;rj + g : 2 2 4 2 @t ; j

(8)

The tensor Pji is the second KCC-invariant of (1). The third, fourth and (fth invariants are:  1 i i i  = (Pj;k − Pk; Rjk  j );   3 i i (9) Bjkl = Rjk;l ;     i Djkl = g;i j;k;l :

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The main result of KCC-theory is the following: Two systems of the form (1) on are equivalent relative to (2) if and only if the :ve KCC-invariants are equivalent. In J x; J˙ t) all vanish if and only if particular, there exists coordinates (x) J for which the gi (x; all KCC-invariants are zero. Let us now introduce the notion of a n-dimensional Finsler space as a manifold where, given a coordinate system (x) and a curve xi = xi (t), the norm of a tangent vector x˙i to the curve at each point P = xi (t) is given by the positive metric function F, |x˙i | = F(x; x), ˙ where F is positively homogeneous of degree 1 in x˙i . From F, a ˙ = (@2 F 2 =@x˙i @x˙j )=2, which must be regular in an metric tensor is de:ned as gij (x; x) open region of the tangent bundle, the collection of all tangent vectors to the manifold. The Berwald’s Gaussian curvature K for two-dimensional Finsler spaces is de:ned from his famous formula [2] i Rjk = FKmi (lj mk − lk mj );

(10)

i where Rjk is given by the :rst equation in (9), li = x˙i =F is the unit vector in the x˙i direction, and mi the unique (up to orientation) unit vector perpendicular to li . Lower index mi is obtained from the unit norm F(x; m) = gij (x; x)m ˙ i mj ≡ mi mi = 1.

3. Construction of the models The symbol N i denotes the number or density of the ith bacterial population, which we assume to satisfy classical logistic dynamics dN i (11) = N i (1 − (i) N i ); dt where we have invoked the pre-symbiont condition, i = . The special notation, as in (i) above, indicates when repeated upper and lower indexes are not to be summed. Because our model will describe ecology and chemical production (in the form of modular bits of RNA), we will use the theory of Volterra–Hamilton systems and introduce the Volterra production equation d xi (12) = k(i) N i : dt This equation embodies a basic assumption, namely, that each population produces one kind of chemical, and each accumulates over time. However, since some of the chemicals may be lost in surrounding media during the information exchange between populations, it is convenient to interpret xi as allometrically related to the total biomass, xi = (i) ln mi , where 0 ¡ i ¡ 1 [8,11,16], so that the quantity (1 − (i) )ln mi represents what is lost in the process. We wish now to ask the question: can n pre-symbiont bacterial populations living according to logistic type Volterra–Hamilton equations (11) and (12) above “evolve” to the status of a symbiocosm, in the sense of Nardon, as refered to in the Introduction? In other words, can we model evolutionary change from the above Volterra–Hamilton system to one that is stable and integrated by information exchange? Furthermore, the “evolved” system must exhibit division of labour, or its ergonomic equivalent, in


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that total energy or cost of chemical production is eQciently allocated between the constituent populations. Since the intention is to model Margulis’ and Woese’s theories in the same mathematical framework, our modelling splits naturally into 2 parts. For the Margulis’ theory, Serial Endosymbiosis, the symbiocosm is internal, while for Woese’s Ancestral Commune theory, it is external and has an intrinsic ecological underpinning. This point can be better understood by recalling the 19th century Russian botanists’ work on lichens [9], external symbioses between algae and fungi, which they showed evolve from loose (ecological) to highly integrated (physiological) interactions. In the Serial Endosymbiosis theory n = 2 and we will allow only the logistic pair interactions. This preserves the form of the logistic Volterra–Hamilton system, (11) and (12). So, in modelling Margulis’ evolutionary process, that form will not change. However, we will allow information xi to enter into the carrying capacity coeQcients i . On the other hand, for Woese n is large and the coeQcients must remain constants, because, in his theory, the commune must be a loosely knit conglomerate of diverse bacterial species. Here, we are interpreting such a conglomerate to be like the early evolutionary stages of the algae-fungi interactions in lichens, that is, weak, non-integrated, ecological interactions. Eqs. (11) and (12) above can be conveniently rewritten in terms of the production parameter s, where ds = et dt, 2 (i) d xi d 2 xi + = 0: (13) ds2 k(i) ds We require that, for either model, our “evolved” system has the form j k d 2 xi i dx dx + Gjk = 0; 2 ds ds ds

(14)

where the n3 coeQcients are constants or involve xi . This class (14) of dynamical systems will encompass both the “primeval” and the “evolved” systems, and serve for modelling either Margulis’ or Woese’ theories. In order to accommodate our ergonomics, i.e., division of labour, we require the production parameter to be given by the cost of production functional, ds = F(x; d x) ¿ 0. Moreover, F is to be positively homogeneous of degree 1 in d x = (d x1 ; : : : ; d xn ), that is, for any positive constant c, dx dx = c F x; ; (15) F x; c dt dt so that ds=dt, the rate of production in the symbiocosm, depends on t individual bacterial rates d xi =dt through the cost F(x; d x=dt). The arc-length s = t12 F(x; d x=dt) dt represents the total production of the symbiocosm in the interval (t2 −t1 ) along a given curve x(t) = (x1 (t); : : : ; xn (t)). The homogeneity of F means that, if all the individual rates d xi =dt are magni:ed by a factor of c, then ds=dt is so magni:ed. We have to introduce the expression Hs = (1=2)F 2 (x; d x=dt), for which (14) are the Euler–Lagrange equations. Note that Hs is therefore positively homogeneous of

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degree 2 in d x=dt, thus, multiplying each d xi =dt by a positive constant c implies that Hs is multiplied by c2 . Furthermore, Hs de:nes two classes of systems, namely, the Riemannian class, where Hs is quadratic in d x=dt, and the Finsler (non-Riemannian) class, where Hs is not quadratic, but is homogeneous of degree 2 in d x=dt. Here are two examples, one for each class. For the former, quadratic case, we may have 2 n 2

d x1 dx 1 i ; (16) Hs = exp(2i x ) + ··· + dt dt 2 while for the latter, non-quadratic but homogeneous case, we may have Hs =

1 (d x2 =dt)2+2= ; exp(2 (x)) 2 (d x1 =dt)2=

(17)

where we have taken n=2, is a positive constant and (x) is an arbitrary polynomial on x1 and x2 . We will see in the following sections that (16) applies to Woese’s theory, while (17) applies to Margulis’ theory. 4. The endosymbiosis model Starting from 1 2 2 2

1 d x dx Hs = exp((21 =k1 ) x1 ) ; + exp((22 =k2 ) x2 ) 2 ds ds

(18)

we get (13) as Euler–Lagrange equations. In fact, these are the geodesic equations of the Riemannian metric ds2 = exp((21 =k1 ) x1 ) (d x1 )2 + exp((22 =k2 ) x2 ) (d x2 )2 ;

(19)

its corresponding arc-length s being a measure of the total production. Furthermore, a smooth coordinate transformation (x1 ; x2 ) → (u1 ; u2 ), where u1 =

k1 exp((1 =k1 ) x1 ); 1

u2 =

k2 exp((2 =k2 ) x2 ); 2

(20)

converts (19) into the Euclidean metric ds2 = (du1 )2 + (du2 )2 ;

(21)

where s in (19) and (21) are the same. The Gaussian curvature [12] of this metric is zero, and it follows from a theorem of Jacobi [12] that solutions trajectories of (13) are Liapunov unstable. Note that Jacobi’s theorem about Riemannian geodesics answers the stability of production question for the Volterra–Hamilton systems (11), (12). Indeed, being straight lines, solutions with close initial values (x0i ; (d xi =dt)0 ) will not remain close generally in x-space, the system being, therefore, unstable.


749

Now we ask if it is possible to modify the expression for Hs so that it leads to Euler–Lagrange equations of the form: 1 2 d 2 x1 dx 1 = 0; + + "1 2 ds k1 ds 2 2 d 2 x2 dx 2 + + "2 = 0; (22) 2 ds k2 ds where "i are smooth functions of (x1 ; x2 ). It is natural to consider 1 2 2 2

dx dx 1 exp(2 1 ) ; + exp(2 2 ) Hs = ds ds 2

(23)

where 1 = #1 x1 + $(x1 ; x2 ) and 2 = #2 x2 + (x1 ; x2 ), and #i = i =ki , as a perturbation of the original system, which leads to the Euler–Lagrange equations dy1 + (#1 + ;1 )(y1 )2 + 2(;2 )y1 y2 − $;1 exp(2( ds

2

−

1 ))(y

dy2 + (#2 + $;2 )(y2 )2 + 2($;1 )y1 y2 − ;2 exp(2( ds

1

−

2 ))(y

2 2

) = 0;

1 2

) = 0;

(24)

where yi = d xi =ds. In order that this system to be of the form (23), we must have $;1 =0 and ;2 =0, so that each i is only a function of xi ; i = 1; 2. It follows that the curvature K (10) vanishes, because, by changing to coordinates (u1 ; u2 ) given by du1 = exp(

1 (x

1

) d x1 );

du2 = exp(

2 (x

2

) d x2 );

(25)

we get 1 Hs = [(du1 )2 + (du2 )2 ]: 2

(26)

Thus, our simple approach is insuQcient to obtain a model with the above stated properties. To solve the problem we need to use the techniques of Finsler geometry [2]. Our main result is that, for Hs given as in (17), with (x)=−1 x1 +(+1) 2 x2 + 3 x1 x2 , and ¿ 0, i ¿ 0 and 3 = 0, the Euler–Lagrange equations are dy1 + (1 − 3 x2 )(y1 )2 = 0; ds dy2 3 1 + 2 + x (y2 )2 = 0: ds +1

(27)

Note that, if 3 = 0, then the original double logistic system (13) are obtained. Moreover, Liapunov stability of this system is completely determined by the sign of the curvature (10): 1 1+2= y 2 3 exp(−2[ − 1 x1 + ( + 1)2 x2 + 3 x1 x2 ]): (28) K= +1 y2

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If 3 ¿ 0, then stability results, while the reverse is true for 3 6 0. Geodesics of two-dimension positively curved spaces, such as a sphere, will remain close generally in x-space, the system being, therefore, stable, while for those with a negative or zero curvature, as a trumphet or a plane, respectively, will not, yielding unstable systems. The parameter 3 is called the exchange parameter. 5. The ancestral commune model In this model we disallow explicit xi in the coeQcients, but allow the number of species to be large. This is our model of a “loose conglomerate of diverse bacterial species”. Neither do we allow the coeQcients to depend on the populations sizes, as this would be inclusion of social interactions [1,2]. As before, we will try for a quadratic cost functional as the simplest possible. Using the fundamental theorem of Volterra–Hamilton systems [1,2] to arrive at (16) above as the n-dimensional functional, using the additional assumption that our community is simple. This means precisely that each bacterial species in the conglomerate exchanges chemical information with at least one other. If this were not assumed, then the fundamental theorem ensures that the conglomerate splits into simple sub-communities, each of the above type, i.e., having (16) as cost functional, with n varying from one community to another, totalling the original number of species. Eq. (14), which is generated by the cost functional (16) through the calculus of i variations, are actually Euler–Lagrange equations, with coeQcients Gjk as i = 0; Giii = i ; Gjk

Giji = Gjii = j ; Gjji = −i ;

i = j = k i = j

i = j:

(29)

These coeQcients of interaction completely characterize our model of Ancestral Comi , with all indexes di8erent, vanishes. For mune. It is especially signi:cant that the Gjk 1 2 3 ; G12 all vanish, which means there example, for a 3 species conglomerate, G23 ; G13 are no higher-order interactions, so that species 2 and 3 do not have an interaction which inTuences species 1, etc. This is a consequence of our model. In order to determine the Liapunov stability for this system, we can analyse the expression for the sectional curvature K(u; v), the Gauss curvature of a surface spanned by the vectors ui and vi . Here, we have obtained such expression taking one of the vectors to be the tangent unit vector li = yi =F, keeping the second arbitrary. We have made use of the Finsler [17] computer algebra package, based on Maple [7]. The maple session is reproduced below: ¿ read ‘d:/:nsler7/riemann.mpl’: ¿ read ‘d:/:nsler7/:nsler.mpl’: ¿ Dimension := 3: ¿ coordinates(x1; x2; x3): ¿ Dcoordinates(d x1; d x2; d x3):


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The coordinates are : X

1

= x1

X

2

= x2

X

3

= x3

The d-coordinates are : Y

1

= d x1

Y

2

= d x2

Y

3

= d x3

¿ F2 := exp(2∗ a1∗ x1 + 2∗ a2∗ x2 + 2∗ a3∗ x3)∗ (d x12 + d x22 + d x32 ); F2 := e(2a1x1+2a2x2+2a3x3) (d x12 + d x22 + d x32 ) ¿ metricfunction(F2) : The components of the metric are : g

x1 x1

= e(2a1x1+2a2x2+2a3x3)

g

x2 x2

= e(2a1x1+2a2x2+2a3x3)

g

x3 x3

= e(2a1x1+2a2x2+2a3x3)

¿ K := factor (simplify(eval(K(v1; v2; v3)))); K := −e(−2a1x1−2a2x2−2a3x3) (a3 d x2v1 − v3a1 d x2 − v2a3d x1 + v2a1d x3 + v3a2d x1 − a2d x3v1)2 =(v12 d x22 + v12 d x32 − 2v1v2d x2d x1 − 2v1v3d x3d x1 + v22 d x12 + v22 d x32 − 2v2v3d x3d x2 + v32 d x12 + v32 d x22 ) The expression in the denominator is the well-known element of area, being always positive, and therefore we can see that K(l; v) 6 0, and therefore Eqs. (14) are unstable, i.e., chemical exchange processes (whole trajectories) for the Ancestral Commune are unstable and aperiodic. Another consequence is that the scalar curvature R [12] is given by R = −(n − 1)(n − 2)exp(−2i xi )[(1 )2 + · · · + (n )2 ]:

(30)

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As one can notice from the above equation, R increases quadratically with n (all other quantities being held constant), becoming ever more negative, and therefore unstable, the larger the commune. The Riemann scalar curvature R and Berwald’s Gaussian curvature K play an important role in stochastic problems in curved spaces [3]. 6. Conclusion Two modern theories of biological evolution, one by Carl Woese, and the other by Lynn Margulis, are modelled with Volterra–Hamilton systems and compared. It was found that the chemical information exchange process in the Margulis’ model, involving only two species of bacteria, is Jacobi stable, while Woese’s model, involving necessarily large number of loosely interacting species, is Jacobi unstable. Therefore it would seem that Woese’s theory su8ers, because stable chemical exchange is needed to set the stage for evolution to occur. In fact, this conclusion is a direct consequence of our interpretation of a “loose conglomerate of diverse bacterial species” as classical ecological interactions, i.e, constant connection coeQcients. Of course, it is possible, in Woese’s theory, for any two species to exchange chemical information, as in Margulis’ theory, after the prescribed evolutionary e8ects have taken place, but then, it is no longer Woese’s theory. On the other hand, Margulis’ theory does not allow for more than a few species to interact at the same time. In conclusion, modelling would seem to indicate that a combination of both theories would be a better evolutionary strategy than either separately. Acknowledgements PLA was partially supported by NSERC grant A-7667 and by LNCC/Rio de JaneiroBrazil during his stay as a visitor from May to July 2001. SFR acknowledges the :nancial support of the Brazilian research support agency CNPq. References [1] P. Antonelli, R. Bradbury, Volterra–Hamilton Models in Ecology and Evolution of Colonial Organisms, World Scienti:c, Singapore, 1996. [2] P. Antonelli, R.S. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993. [3] P. Antonelli, T. Zastawniak, Fundamentals of Finslerian Di8usion with Applications, Kluwer Academic Publishers, Dordrecht, 1998. [4] E. Cartan, Observations sur le mVemoire prVecVedent, Math. Z. 37 (1933) 619–622. [5] S.S. Chern, Sur la gVeomVetrie d’un systWeme d’Vequations di8erentielles du second ordre, Bull. Sci. Math. II 63 (1939) 206–212. [6] http://www.nytimes.com/library/national/science/061300sci-life-origins.html. [7] http://maplesoft.com. [8] J. Huxley, Problems of relative growth, 2nd Edition, Dover Press, 1972. [9] L.N. Khakhina, Concepts of Symbiogenesis, Yale University Press, New Haven, 1992. [10] D. Kosambi, Parallelism and path-spaces, Math. Z. 37 (1933) 608–618.

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