It is remarkable that the concept and process of self-organization feature in so many different major disciplines [1]: in Applied Mathematics (Robotics, pioneered by von Neumann [2] and also [3] ), in Physics (by a thermodynamic viewpoint, e.g. [4] ) in Chemistry (primarily Organic Chemistry, and specifically Dynamic Combinatorial Chemistry [5] [6] and also in Crystallography) and, notably, in Biology (e.g., DNA copying as in meiosis, mitosis or binary fission, in cell divisions [7] [8] [9] ). With a net so widely spread, it cannot be expected to be doing justice to all the work done heretofore.

While addressing the subject with methods in physics, the inquiry takes us to Biology, and particularly to a specific phenomenon of self-organization, namely, self-replication (SR) [6], existing in a large variety of biological entities and recognized as one of the main marks distinguishing animate from non-animate matter. (Though, there may be rare instances of SR also in the inorganic world, but not as a cyclic process [10], or in auto-catalysis but lacking the specificity of SR [11].) Notably in Biology, SR occurs at enormously differing size scales, ranging from large animals to tiny biomolecules. Accounts of real life and laboratory occurrences of SR can be found in [11] [12], including experiments at synthesising self-replication by Leslie Orgel, Rebek, von Kiedrowski [13] [14] [15] and others, as well as template-guided nonenzymatic SR, the latter also characterizing primordial RNA replication [16]. Past theoretical works, including Stuart Kauffman’s set autocatalysis, have been summarized in [17] [18], with a mathematical framework that addressed in particular the citric acid - >CO and the formose (formaldehyde - > sugar) reactions.

We have made it our aim to connect up biological SR with Physics and that at a most basic level. The price that we pay for the temerity of such undertaking is doing the job in an extreme schematic and simplified fashion, thereby producing only a preliminary investigation, to be hopefully followed by further more specific and detailed works.

Remarkably enough, the guiding idea of the present approach (though arrived at independently by us) had already been given expression in a work remote from basic Physics, in a review of Dynamic Combinatorial Chemistry (DCC) dealing with the autonomous formation of receptors, that read: “Addition of a guest (a template molecule in the terminology of the present study, RE), which binds strongly (my italics) to one of the libraries (available components, RE) shifts the equilibrium towards its formation (of a new creature or offspring, RE), resulting in the amplification (renewed formation, RE) of the successful receptor at the expense of the less successful” [5]. The italicised words “binds strongly” are the leitmotif of the present work, as will be apparent in the sequel. Even more supportive is the concluding sentence of the same reference: “We are confident therefore that DCC will engender generations of ingenious concepts and lead to new, as yet unpredictable theories”. Yet closer to the approach here taken is the assertion made about 70 years earlier than that: “The living organism seems to be a macroscopic system which in part of its behaviout approaches to that purely mechanical (as contrasted with thermodyamic) conduct to which all systems tend, as the temperature approaches absolute zero …” ( [19], Chapter 6). It seems that we are at liberty, and not in conflict with the author, to qualify “mechanical” by quantum mechanical, as will be argued next. Regarding the quantum mode of the formalism, this is analogous to Kramers’ and P.W. Anderson’s mechanism for superexchange [20] [21], with the bath replacing the bridging anion and exploitation of the randomness of the former. [Note the “attractive coupling” expression (based there on third order perturbation) of Anderson following his Equation (19) in [21].]

At the foundation of all forms (situations, states, phenomenologies) and processes (changes, developments, steady existence) that constitute our daily life lies Quantum Mechanics (QM), mostly expressed in some or other form of the Schrödinger equation, erstwhile involving a Hamiltonian and some boundary initial conditions. In most complex issues where the QM path is not feasible, Thermodynamics (Td) or Statistical Mechanics comes to the rescue and one must hope that they (QM and Td) are compatible, as seems to be the case. The merging of the two, essentially in the direction of QM - >Td, has become a major recent objective of research, going under the name of Eigenstate Thermalization Hypothesis [22] [23].

Life (and more precisely, biological processes), though an extraordinary and unique phenomenon, ought not to be an exception to a derivation of its thermodynamics from quantum theory, as has been the case in e.g., light production or lasing. By my understanding, a quantitatively specific description of life starting at the QM level does not exist, not even within the broad insights provided by Schrödinger [19], Bernal [24], Prigogine [25], Dyson [26] and Ruelle [27], among others.

In the following, the subject will be self-replication (SR) in biological systems, a process which is both ubiquitous and perennial. The systems in view range in their shape and size from animals (humans included) through cells to biological micro-molecules; from a Physics viewpoint every single entity in a particular system is characterized by its momentary state, inclusive of its eigen-state, orientation and location. In the following schematic and maximally simplified treatment all these varied systems are represented in one single model and all their properties are subsumed under a single yes-no alternative. (A classical, non-quantal description might lead to qualitatively similar results, but without the sharpness consequent upon quantization, e.g., of spin alignment.) Of late, SR was treated by J. England [28] [29] from a thermodynamic viewpoint. Though Ruelle [30] has formalized that work in a quantum setting, the subject still awaits a microscopic model, like the one proposed here.

The conceptual innovation of this work lies in that, contrasting to the currently prevalent understanding of replicative and other processes in presently existing living systems, as due to the background presence of enzymes that lower and raise barriers to initiate and perpetuate modifications (Figure 3 in [28] and P. Gaspard’s remark in [30] ), but whose dynamical variables are absent from the formalism, these are here included in an essential, though only schematic way. It should be emphasized that this work aims at providing a descriptive account of how SR emerges in a quantum mechanical setting, rather than why the settings occurs in systems that exhibit SR. Specifics are in section 4.1.

The article starts in section 2 with a list of the dramatic personae which enumerates the entities that take part in a real self replication event, a list that is followed by another, section 2.1, in which each biological component is schematically represented by a qubit (half-spin). The enumeration of components matches, but is more specific than those described by Ruelle [30]. (e.g., his “X” are here sub-listed as T, R, L/(T, R).) In subsection 3 the present approach is placed in a perspective of standard scenarios of enzymatically activated biologocical activities, pinpointing how the former is complementary to the latter.

The model Hamiltonian is given in section 4, followed by a curt but important derivation of the “control” term in subsection 4.1. The entropic state of the system is given in section 5, both before (subsection 5.1) and at the conclusion (subsection 5.2), of the replication process. The numerical results are displayed in section 6 with the simple Figure 4.

As a side line, this work points tentatively, and on the basis of the results, at a partial answer to the quintessential question “what discriminates between animate and non-animate matter?” as contained in section 6.1. (It should be stressed that the criteria derived from this work are not in opposition to general characterisations of life, [19] - [26], but give them here an expression within a basic, Hamiltonian formalism.) Subsection 6.2 elucidates the energy scales used, followed by a Conclusion section 7. The Supplementary Information (SI) informs how the tracing in a highly entangled system is to be performed. The theorems there may have a wide application.

1) The template (T), which duplicates and maintains its state during SR.

2) The library (L) of available molecules, which is composed of

a) the template’s counterparts, the reactants (designated r 1 , r 2 , ⋯ ), i.e., those molecules that are gathered up by T from just anywhere in the library ultimately to be the replicate “R”, and

b) those that are unselected, but may play some role (e.g., as enzymes) in SR. These do not replicate (by the “Central Dogma” of biology).

3) The broader environment, which adjusts in the course of replication and constitutes a thermal bath supplying the needed free energy [7].

This section examines the relation of the present quantum state formalism in an SR process to other scenarios in which SR arises as an enzyme coordinated barrier jumping event. Several theories for barrier jumping in enzymatic actions are described in [31], which also amplifies the common formulation of the process along a single reaction coordinate mode by the addition of a “promoting” vibrational mode to form a two dimensional landscape of the potential energy surface. Interestingly, a “promoting mode” of opposite symmetry to accepting modes entered previously the theory of vibronic (vibrational-electronic) decay in large molecules [32].

The following figure (Figure 1) serves to visualise the complementary roles played by the two schemata for SR as transitions between two parabolic Potential Energy Surfaces (PES), whose profiles are drawn along a single reaction coordinate. The simple parabolic depiction of the biological entities does not do justice to the complexity and multiplicity of sub-conformational wells in real systems, but serves as a base. In Figure 1(a), PES are drawn on the left (in black) for the pre-SR configuration, template T and pre-replicated reactants r 1 + r 2 + ⋯ , and on the right (in red) for the post-SR configuration, template T and the replicate R in which the reactants have merged, with a separation barrier in-between. To initiate a left-to-right transition, the barrier is transiently lowered, shown by the short curly line and a promoting coupling between the configurations (not shown) is brought into play. This summarizes the barrier jumping scenario of SR, in which the quantum states involved both in the ascent to the transition state and in the descent from it are not specified.

In the adjacent Figure 1(b), illustrative of the present, quantum state scheme, PES are plotted, for the situations in which the Hamiltonian is without a control term (W = 0, upper drawing) and one with a large control ( ≫ 1 , lower drawing).

For each Hamiltonian alternative configurations ( T + r 1 + r 2 + ⋯ ) and (T + R) exists (and are drawn in their respective colouring, as before), but their low energy states (indicated by the short horizontal curves at the bottoms of the parabolae) change radically. (The two alternative configurations are formalized in our two spin model by ↑ ↓ and ↑ ↑ , respectively, also shown in the drawings.) The source of the energy change is the increase of the control term coefficient W in the Hamiltonian from 0 to a large value. (The physical cause of this change is not clear; it is presumed to have an enzymatic source.) Whereas at low temperatures the resultant configuration is both determined and described by the ground state, the interim quantum states are not part in the theory. Their exclusion from the theory is indicated by the shaded regions in the figure at barrier heights: the events in this region taking place during SR are better viewed within the barrier jumping scenario.

The control W, which is thus the pivotal source of the present complementary scheme, is derived by a straightforward quantum mechanical procedure in subsection 4.1.

The foregoing processes are entirely rooted in the following Hamiltonian (written in arbitrary energy units)

H = H ( N ) + H b a t h + δ H N − b a t h (1)

in which the first term for N interacting spins (described immediately below) is fully treated, the second term, the bath Hamiltonian is not required for our considerations , while the third, interaction term is dealt with perturbationally in the following subsection 4.1. Within this model of N fully interacting 1/2-spins (qubits), the spins labelled 1, ⋯ , N − 2 are identified with the background items 2b’ in the above enumeration, (these do not replicate and are not directly subject to SR while being part of the “library”); next, a spin labelled N − 1 which is the template (“T”, 1’) in the above and, finally, labelled N (and designated “R” for “replicate”, 2a’) is the spin that is being converted in SR to become the facsimile of the template. (The peculiar ordering of the labels aims at a simplified formulation of the “reservoir tracing”, which is the concern of the Supplementary Information.) The following Hamiltonian applies to this spin system:

H ( N ) = − ∑ i = 1 , ⋯ , N     h i σ z i − ∑ i , j = 1 , ⋯ , N     g i , j σ x i σ x j (2)

The Zeeman energy parameters differ in their magnitudes, as follows:

i = 1 , ⋯ , N − 2 : h i = O ( 1 ) ;     i = N − 1 : h T ≫ 1 ;       i = N : h R = O ( 1 ) (3)

For the inter-spin coupling we have chosen

g i j = O ( 1 )     ( i , j = 1 , ⋯ , N ) (4)

Further details on the chosen value of the coupling coefficients are found in [33]. These are not here reproduced, as these details are not essential to the following for which the important features in the Hamiltonian are the template binding term − h T σ z T and a correction term hereby derived.

For δ H N − b a t h we postulate a bilinear form:

δ H = ∑ b ( k b T σ b ∗ σ T + k b R σ b ∗ σ R ) (5)

with T and R labelling the template and replicate spin-entities and b indexing the bath qubits. * denotes vectorial inner product. Calculating its perturbational effect on the energy of the combined T-R-bath wave-function Ψ , one may assume that due to the random nature of the bath the first order diagonal perturbational correction 〈 Ψ | δ H | Ψ 〉 vanishes. Regarding the second order energy correction, we first of all note that for low lying states the energy correction is predominantly negative. Next we perform a number of manipulations (approximations) that will end up by supplying an addition to the Hamiltonian H ( N ) above of the form − W ′ σ z T σ z R , with W ′ large and negative.

δ ( 2 ) = Σ e x c | 〈 Ψ | δ H | Ψ e x c 〉 | 2 E − E e x c (6)

≈ − | δ E | − 1 Σ e x c ( 〈 Ψ | δ H | Ψ e x c 〉 ) 2 (7)

≈ − | δ E | − 1 〈 Ψ | ( δ H ) 2 | Ψ 〉 (8)

= − | δ E | − 1 Σ b 〈 〈 Ψ | [ k b T σ b ∗ σ T ] 2 + [ k b R σ b ∗ σ R ] 2       + 2 k b T k b R σ b ∗ σ T ∗ σ b ∗ σ R | Ψ 〉 〉 (9)

In passing from the first to the second line all energy denominators involving excitational energies E e x c were approximatively replaced by a constant and negative mean value, thus enabling by the closure property to reach the next stage, in which appears the expectation value of ( δ H ) 2 . Opening this, we are led to a summation over bath entities b. At this stage, cross-terms in the bath indices were deemed to vanish by the randomness of phases in the bath spins. Subsequently, after opening up ( δ H ) 2 in line four, the two squared terms contribute only constants to the energy and may be ignored. We are thus left with the cross term coupling target and replicate spins bi-linearly. Assuming identical signs for the coupling coefficients k b T and k b R , each term in the last expectation sum contributes negatively. Adopting the so named “Effective Hamiltonian” procedure in electron spin resonance (ESR) theory, the operator between the bra-kets can be appended to the Hamiltonian H ( N ) in Equation (2) as − W ′ σ z T ⋅ σ z R with W ′ given by

W ′ = 2 | δ E | Σ b k b T k b R > 0 (10)

The magnitude of W ′ is a matter of further investigation and modelling of the bath, but it may be noted that it is a sum of many terms, in principle numbering the molecular contents of the bath. There may be analogous terms, involving x and y -spin components which are non-diagonal and presumably play a role in the energy exchange between the bath and the T-R system. They have no role in the model.

Summarizing the foregoings, we have identified within the T-R-bath-system Hamiltonian a potential source for the strongly negative bilinear coupling of the form

− W σ z T ⋅ σ z R (11)

in which W = W ′ + g T R is the control parameter for self-replication.

The above approximative procedure of eliminating the bath variables due to their random (and uncorrelated) phases appears to be justified at the present level of treatment, in which the bath degrees of freedom are not dynamic (i.e. not subject to equations of motion; alternative justifications rooted in the sheer size of the number of bath coordinates were also given). [In a more elaborate treatment inclusive of bath dynamics the foregoing manipulation of ( δ H ) 2 would have been done in reverse: by tracing over the N-variables (the total system minus the bath) and obtaining an “Effective Hamiltonian” representing interaction within the bath coordinates, inducing correlations and entropy increase, for which predictions have already been made in [34] [35]. By their estimate, this occurs mainly well after the SR process, thus allowing one to treat the process of SR itself at the level here adopted.]

In this conceptual stages are meant, as in a “gedanken experiment”, not chronological ones. For the latter, the states of bifurcation, appearing in Figure 4 as a diagonal line, may well be relevant, representing states of meta-stability.

The numerical interplay between the control parameter W of the bilinear T-R term and the “binding energy”, linear, Zeeman term strength parameter h_R of the replicate is displayed in Figure 4. In this the reduced density of the replicate is contour-plotted with W the abscissa and h_R the ordinate. The replicate density

is strongly demarcated, with the light blue region (in the right-lower part) representing alignment (replicating capability) between R and T and dark blue region (in the left-upper part) indicating anti-alignment, in the lowest energy state. From the graph, formally for the coefficient W entering the Hamiltonian as − W σ z T σ z R ,

W >   h R (16)

is the requirement for alignment, a criterion that is clearly favoring a strongly attractive control. Below this line the upper state density of the replicate tends to be unity alike to the template, while above the line it tends to vanish. The diagonal is a bifurcation line between the aligning and anti-aligning tendency of the ground state and on it the two lowest lying states with opposing alignment tendencies are energetically co-degenerate, implying an entropy increase. (The zig-zag nature of this line is a numerical artefact.) In the immediate vicinity of the line, the reduced replicate density is not yet close to 1 or 0, this is only the case further inside the respective regions. Speculatively, one may associate the bifurcation line with an inherent instability (“criticality” in Statistical Physics terminology), encountered in some, and perhaps several, biological systems.

The relatively sharp transition between alignment and anti-alignment is a quantum effect; in a classical description the control term would only gradually affect the statistic distributions of the spin orientation. Whereas the alignment changes dramatically across SR, the weights (RDM values) of the constituents in 2b’ vary only minimally by about 5 percent, a property in line with their presumed enzymatic status.

The process of living systems with regard to their self-reproductive capacity has here been given a quantum description, differing from (but not replacing) those current descriptions which present them as processes in a thermodynamically non-equilibrium setting. Admittedly, the actual complexity of biological reality has here been drastically reduced by representation of the manifold components through a single spin or qubit and the multiple properties by a single spin direction. It behoves us (and perhaps others) to extend the treatment to models comprising more entities and a variety of properties. A candidate for future treatment might be the self-assembly of a triatomic molecule, acting in some configuration as a template and inducing the assembly of a chosen trio of atoms selected from a large reservoir of atoms.

It may be objected, though, that by identifying the post-SR state with the ground state due to equilibration, thermalization creeps in, somehow which is external to the strictly Hamiltonian formalism of this work. However, according to the Eigenstate Thermalization Hypothesis in some of its formulations [22] [23], the expectation values of the spin alignment (a local observable) will occur autonomously, without thermal contact with an external heat-bath.

I thank helpful suggestions by Professors I. Bersuker, E. Eisenberg, J. P. Klinman and V. Polinger. The help of Prof. A. Yahalom in writing for our joint paper [33] a Mathematica program of the spin-spin Hamiltonian, and here used, and the technical help of N. and T. Kayser are acknowledged.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The author declares no conflicts of interest regarding the publication of this paper.

Englman, R. (2021) A Quantum State Scenario for Biological Self-Replication. Open Journal of Biophysics, 11, 159-176. https://doi.org/10.4236/ojbiphy.2021.112005

In a previous publication [33] a convenient labelling of the multiple 1/2-spin states was achieved by replacing the ordinary, decimal numbering (in which, e.g., 2⁵ is written as 32) with a binary representation, in which the same number is written as [10000] or possibly [000010000], etc. In this representation, the state of the r-spin is conveniently registered by having 0 or 1 at the r-th position from the right for, respectively the down- or up-state of the state. (While not relevant to the present work we, remark, that for entities or degrees of motion, that have Y rather than two alternative states, with Y being any arbitrary and finite integer, the same idea applies and the combined state would conveniently be represented by an Y-nary representation. Thus, for the not uncommon duodecimal representation, each degree of freedom would have entries at each position any one of the figures 1,2, ⋯ ,10, X , Y . That this is not the end of the story, arises from the fact that in most (if not all) programming languages a summation or repeated operation command refers to the decimal representation and this necessitates a shunting between the representations.)

Here we recall that “tracing” is used in the statistical interpretation of quantum mechanics to obtain expectation values for a local operator (meaning one that operates on only one or a few degrees of motion), while summing over all states in the other degrees of motion. Insomuch as tracing also involves a summation [command], it is of interest to have at one’s disposal a formula that carries out the tracing for states written in the binary (or Y-nary) representations. The following rule is given for the tracing over a single degree of motion’s (namely, the r’th spin’s) density of state (also known as the reduced density of state), but other traces follow from this. Astonishingly, the rule is very simple!

Eigen-solutions of the many-spin Hamiltonian matrix in Equation (2) of the text are of the form

| J 〉 = ∑ n     C n J | n 〉 ,   J = 0 , ⋯ , 2 N − 1 (18)

where n runs over the 2^N combinations of ( 0,1 ) (in that order).

Theorem 1

T r r | 0 〉 〈 0 | r = ∑ J ′   | C n J ′ | 2 ,   J ′ = J     if   J   mod   2 r   = ( 0 , ⋯ , 2 r − 1 − 1 ) T r r | 1 〉 〈 1 | r = ∑ J ″   | C n J ″ | 2 ,   J " = J     if   J   mod   2 r   = ( 2 r − 1 , ⋯ , 2 r − 1 ) (19)

Traces for other local operators than the density operator are treated analogously, with local expectation values inserted in the sums of Equation (19).

We next formulate the problem in more general terms, as an algorithmic exercise, and give a solution in Theorem 2.

N and Y are finite integers. Expressions for the numbers J = 0, ⋯ , Y N − 1 in a Y-nary system of digits, in a correspondence with a binary or decimal system, can be written as

J = { I N   I N − 1 ⋯ I r ⋯ I 1 } ,     I r = 0 , ⋯ , ( Y − 1 ) ,     forall   r = 1 , ⋯ , N (20)

(please note that r = 1 is rightmost and r = N is leftmost). (Example with N = 4, Y = 3: 11 = { 0102 } .) The problem is to seek a rule as to which J’s to include in the trace-sum for a given I r = X , i.e. which J’s contain X at r position, with whatever entries at other positions.

Theorem 2

(Solution of the problem, expressed as a constructive procedure.)

For position r   ( = 1 , ⋯ , N ) arrange all J ( 0, ⋯ , Y N − 1 ) ’s in bunches of size Y r − 1 numbered 1, ⋯ , Y N − r + 1 The trace for any I r = ( 0, ⋯ , Y − 1 ) is then over all J values, starting at the I r + 1 ’th bunch and continuing to other bunches skipping over them Y − 1 -times. (E.g., for Y = 3 , I r = 4 include all J’s in the fifth, eights, etc. bunch)

Proof

When I r first appears J = I r ∗ Y r − 1 (thus in the binary system { 01000 } = 8 = 1 ∗ 2 4 − 1 ) and likewise, when I r + 1 first replaces I r , J = ( I r + 1 ) ∗ Y r − 1 , the difference is Y r − 1 and this is the length of the bunch between them, such that I r is at position r. This is also the bunch length, when there are non-zero digits to the left of r and then one meets again I r after Y − 1 bunches, those in which the digit at position r differs from I r .

Illustration: In the binary system at position r = 1 even and odd digits alternate, while at the last position r = N the first half of 2 N J ’s have 0’s and the second half has 1’s.

Corollary

In many applications, the number of states differ for different degrees of motion, e.g., in a discretization of radial and angular degrees of motion. Then the bunch length at the r’th position is Y r − 1 Y r − 2 ⋯ Y 1 , with skipping J’s Y r − 1 times over.