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On the divergence of species
BioSystems, 24 (1991) 301-303 Elsevier Scientific Publishers Ireland Ltd.
301
On the divergence of species M.A. Livshits and M.V. Volkenstein Engelhardt Institute of Molecular Biology, Academy of Sciences of the USSR, Moscow 11Y984 (U.S.S.R.) (Received October Sth, 1990)
Speciation may be considered as a hind of non-equilibrium phase transition which is caused by bifurcation at a critical point of instability. It can be shown that multiple branching of an evolutionary tree is highly improbable, since non-degenerate mathematical models describe only binary branchings. Keywords: Speciation; Phase transition; Bifurcation; Evolution,
The only figure in The Origin of Species by Charles Darwin shows both binary and multiple branchings of the evolutionary tree. Although biologists usually draw only the binary divergence both in gradual and punctual evolution, they do it for the sake of simplicity and think that multiple branching is quite possible. However, very general considerations show that multivariant branching is highly improbable. Speciation possesses all the features of nonequilibrium phase transitions both of the first and second kind (Belintsev and Volkenstein, 1977; Ebeling and Feistel, 1982; Volkenstein, 1987; Volkenstein and Livshits, 1990). For instance, binary divergence means the loss of symmetry. Let us consider phylogenetic development in t.~lns of dynamic systems theory. The evolution of taxons can be presented as a slow change in the steady states of a dynamic system. At certain stages of evolution, the parameters of the system attain critical values at which the steady states bifurcate and hence their stability is lost. Figure 1 shows all simple steady state bifurcations. The solid lines designate stable states aud the dashed lines mark unstable states. Otner stable states may also exist in the situations shown in Fig. lb and c in the over-critical region. However, these states are remote and
are not explicitly included in the present consideration. In the most interesting bifurcation of the first type (Fig. la), the loss of stability gives rise to two new stable states, i.e. to a binary divergence. The simplest dynamic model for such a bifurcation is 33=
x(p - x2)
(1)
where c is a parameter with the bifurcational value j.4 = 0. For the system with two parameters
x= x(p -
(a)
x2) + E
tP 1
Fig. 1. Steady state bifurcation diagrams.
0303~2647/91/$03.50 0 1991 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
(2)
II
A degeneracy is always induced by some special conditions which produce an elevated sym metry. Small perturbations in the dynamics are able to break the symmetry and to remove the degeneracy. For instance, the elementary cusps of the system (3) are easily split by small shifts:
I
I
I
I I I
‘i J
x=x&a-
(4)
Fig. 2. Binary divergence diagram after symmetry breakdown.
this type of bifurcation belongs to the so-called cusp behavior surface (Poston and Stewart, 1978). A small shift in the second parameter transforms the bifurcation diagram la into that shown in Fig. 2. The continuous transfer is possible only to ore of the two new stable states, either to the left- or to the right-hand state, depending on the sign of E. This means that the state can be chosen due to the perturbation of the dynamic system rather than to the generation of the appropriate initial conditions. Multivariant branching is possible only in degenerate bifurcation. Figure 3 illustrates such a bifurcation where four new stable states (x = &da, x = ldplc) and two unstable ones (SC= +J&) appear simultaneously at the critical point of instability in the original branch. This bifurcation diagram corresponds to the triple cusp:
\ Y
3c= (p - cd)(p - bxqo(- cx2)+
\
\
\\
\
1
\
I
1
I I
1
I I
CdJ&-P-bXp(p-~-CX2)
E
(3)
After such a splitting of the degenerate bifurcation, the continuous transfer maintaining the stability leads only to two most remote branches, i.e. to two maximally diverse states. Some examples of the corresponding diagrams are shown in Fig. 4. An additional perturbation by the additive parameter e preserves the continuous way to only one of the two stable states. Only one degree of freedom has been considered in these dynamic models. In a more general case, the mutual influence of different degrees of freedom can produce secondary bifurcations of stable branches. This means that the splitting of degeneracy transforms the multivariant branching into successive binary branchings. The simplest model for such a situation is presented by the equations i =
x(e(r- x2 - cg2)
fj = $(/A-
A
- y2 - cs2)
where A is the splitting and cyc 1. The corresponding bifurcation diagram is shown in Fig. 5. Similar though more complex examples of secondary bifurcations in the theory of dissipative structures are considered in the paper by Belintsev et al. (1981).
I
/’
,f
Fig. 3. Triple cusp degenerate bifurcation.
(5)
Fig. 4. Examples of the triple cusp splitting.
Fig. 5. Two-dimensional splitting into consecutive binary b&cations.
These model calculations allow one to conclude that only binary bifurcations occur in the evdutionary trees. A multiple branching picture of the kind shown in Fig. 6a means that the actual diagram must have nearby consecutive bifurcations similar to Fig. 6b. Of course this presentation cannot be considered as a rigorous proof of the impossibility of multiplebranching-sin speciation. However there are important features of speciation which make it possible to treat it as a kind of bifurcation in an open system which is far from equilibrium.As has been shown by Eigen et al. (1989) in much simpler systems of viruses, the extinction of a species and the prerequisites for a new speciation occur as phase transitions due to an instability of the previous situation. The results of our paper show once more the value of studying speciation and macroevolution as bifurcational phenomena.
a
I
0
Fig. 6. Degenerate and split brancbings of the evolutionary tree.
References Belintsev, B.N., Livshits, M.A. and Volkenstein, M.V., 1981, Pattern formation in systems with nonlocal interactions. 2s. Phys. B., Condensed Matter 44.345-351. Belintsev, B.N. and Volkenstein, M.V., 1977, Phase transition in an evolving population. Dokl. Akad. Nauk SSSR 235, 205-210 (in Russian). Ebeling, W. and Feistel, R., 1982, Physik der Selbstorganisation und Evolution (Akademie-Verlag, Berlin). Eigen, M., MC&skill, J. and Schuster, P., 1989, The molecular quasi-species, in: Advances in Chemical Physics, I. Prigogine and S.A. Rice (eds.) Vol. 75, pp. 149-263. Poston, T. and Stewart, I., 1978, Catastrophe Theory and Its Applications (Pitman, London). Volkenstein, M.V., 1987, Punctualism, non-adaptationism, neutralism and evolution. BioSystems 20, 289-304. Volkenstein, M.V. and Livshits, M.A., 1989, Speciation and bifurcations. BioSystems 23, l-5.
301
On the divergence of species M.A. Livshits and M.V. Volkenstein Engelhardt Institute of Molecular Biology, Academy of Sciences of the USSR, Moscow 11Y984 (U.S.S.R.) (Received October Sth, 1990)
Speciation may be considered as a hind of non-equilibrium phase transition which is caused by bifurcation at a critical point of instability. It can be shown that multiple branching of an evolutionary tree is highly improbable, since non-degenerate mathematical models describe only binary branchings. Keywords: Speciation; Phase transition; Bifurcation; Evolution,
The only figure in The Origin of Species by Charles Darwin shows both binary and multiple branchings of the evolutionary tree. Although biologists usually draw only the binary divergence both in gradual and punctual evolution, they do it for the sake of simplicity and think that multiple branching is quite possible. However, very general considerations show that multivariant branching is highly improbable. Speciation possesses all the features of nonequilibrium phase transitions both of the first and second kind (Belintsev and Volkenstein, 1977; Ebeling and Feistel, 1982; Volkenstein, 1987; Volkenstein and Livshits, 1990). For instance, binary divergence means the loss of symmetry. Let us consider phylogenetic development in t.~lns of dynamic systems theory. The evolution of taxons can be presented as a slow change in the steady states of a dynamic system. At certain stages of evolution, the parameters of the system attain critical values at which the steady states bifurcate and hence their stability is lost. Figure 1 shows all simple steady state bifurcations. The solid lines designate stable states aud the dashed lines mark unstable states. Otner stable states may also exist in the situations shown in Fig. lb and c in the over-critical region. However, these states are remote and
are not explicitly included in the present consideration. In the most interesting bifurcation of the first type (Fig. la), the loss of stability gives rise to two new stable states, i.e. to a binary divergence. The simplest dynamic model for such a bifurcation is 33=
x(p - x2)
(1)
where c is a parameter with the bifurcational value j.4 = 0. For the system with two parameters
x= x(p -
(a)
x2) + E
tP 1
Fig. 1. Steady state bifurcation diagrams.
0303~2647/91/$03.50 0 1991 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
(2)
II
A degeneracy is always induced by some special conditions which produce an elevated sym metry. Small perturbations in the dynamics are able to break the symmetry and to remove the degeneracy. For instance, the elementary cusps of the system (3) are easily split by small shifts:
I
I
I
I I I
‘i J
x=x&a-
(4)
Fig. 2. Binary divergence diagram after symmetry breakdown.
this type of bifurcation belongs to the so-called cusp behavior surface (Poston and Stewart, 1978). A small shift in the second parameter transforms the bifurcation diagram la into that shown in Fig. 2. The continuous transfer is possible only to ore of the two new stable states, either to the left- or to the right-hand state, depending on the sign of E. This means that the state can be chosen due to the perturbation of the dynamic system rather than to the generation of the appropriate initial conditions. Multivariant branching is possible only in degenerate bifurcation. Figure 3 illustrates such a bifurcation where four new stable states (x = &da, x = ldplc) and two unstable ones (SC= +J&) appear simultaneously at the critical point of instability in the original branch. This bifurcation diagram corresponds to the triple cusp:
\ Y
3c= (p - cd)(p - bxqo(- cx2)+
\
\
\\
\
1
\
I
1
I I
1
I I
CdJ&-P-bXp(p-~-CX2)
E
(3)
After such a splitting of the degenerate bifurcation, the continuous transfer maintaining the stability leads only to two most remote branches, i.e. to two maximally diverse states. Some examples of the corresponding diagrams are shown in Fig. 4. An additional perturbation by the additive parameter e preserves the continuous way to only one of the two stable states. Only one degree of freedom has been considered in these dynamic models. In a more general case, the mutual influence of different degrees of freedom can produce secondary bifurcations of stable branches. This means that the splitting of degeneracy transforms the multivariant branching into successive binary branchings. The simplest model for such a situation is presented by the equations i =
x(e(r- x2 - cg2)
fj = $(/A-
A
- y2 - cs2)
where A is the splitting and cyc 1. The corresponding bifurcation diagram is shown in Fig. 5. Similar though more complex examples of secondary bifurcations in the theory of dissipative structures are considered in the paper by Belintsev et al. (1981).
I
/’
,f
Fig. 3. Triple cusp degenerate bifurcation.
(5)
Fig. 4. Examples of the triple cusp splitting.
Fig. 5. Two-dimensional splitting into consecutive binary b&cations.
These model calculations allow one to conclude that only binary bifurcations occur in the evdutionary trees. A multiple branching picture of the kind shown in Fig. 6a means that the actual diagram must have nearby consecutive bifurcations similar to Fig. 6b. Of course this presentation cannot be considered as a rigorous proof of the impossibility of multiplebranching-sin speciation. However there are important features of speciation which make it possible to treat it as a kind of bifurcation in an open system which is far from equilibrium.As has been shown by Eigen et al. (1989) in much simpler systems of viruses, the extinction of a species and the prerequisites for a new speciation occur as phase transitions due to an instability of the previous situation. The results of our paper show once more the value of studying speciation and macroevolution as bifurcational phenomena.
a
I
0
Fig. 6. Degenerate and split brancbings of the evolutionary tree.
References Belintsev, B.N., Livshits, M.A. and Volkenstein, M.V., 1981, Pattern formation in systems with nonlocal interactions. 2s. Phys. B., Condensed Matter 44.345-351. Belintsev, B.N. and Volkenstein, M.V., 1977, Phase transition in an evolving population. Dokl. Akad. Nauk SSSR 235, 205-210 (in Russian). Ebeling, W. and Feistel, R., 1982, Physik der Selbstorganisation und Evolution (Akademie-Verlag, Berlin). Eigen, M., MC&skill, J. and Schuster, P., 1989, The molecular quasi-species, in: Advances in Chemical Physics, I. Prigogine and S.A. Rice (eds.) Vol. 75, pp. 149-263. Poston, T. and Stewart, I., 1978, Catastrophe Theory and Its Applications (Pitman, London). Volkenstein, M.V., 1987, Punctualism, non-adaptationism, neutralism and evolution. BioSystems 20, 289-304. Volkenstein, M.V. and Livshits, M.A., 1989, Speciation and bifurcations. BioSystems 23, l-5.