Modeling Genomic Fixation Probabilities of Tautomeric Mutations as a Function of Characteristic Fluctuations in Earth’s Magnetic Field
Postulate 1.0 Degrees of freedom at any locus on a strand of DNA are an intrinsic property of the sequence in which that locus is situated.
Let α be a particular locus on some DNA strand A, and let L be an arbitrary distance spanned by some sequence Sᴀ of A at temperature T, such that Sᴀ contains locus α:
Let n be the minimum number of nucleotides Sᴀ must theoretically contain to span L in the special case where the effects of T are limited to changes in intermolecular bond distances but do not inform the configuration of Sᴀ, with the result that Sᴀ is arranged linearly. Now let M be the combined mass of these n nucleotides in the special case.
Consider that if Sᴀ is divided into N segments of an equal number of nucleotides, and of average mass s, then
where d is the spatial dimension of Sᴀ. Note that in the special case, where the configuration of Sᴀ is linear, d = 1, which is intuitive. Note too that outside of the special case, where T affects configuration, d corresponds to some fractal dimension (Figure 1) of Sᴀ. Therefore we can express the capacity dimension, dc, of Sᴀ as the limit of d as s → 0:
It is observed that dc describes an intrinsic property ofSᴀ related to its nucleotide sequence; that because L is arbitrary, dc must also describe an intrinsic property of α; and that these intrinsic properties can only be determined experimentally. Moreover, it is observed that over time the degrees of freedom at α increase as the capacity dimension increases, and decreases as the capacity dimension decreases. Let us then define this relationship¹ as the function R₁, such that
where dα(t) is the degrees of freedom at α as a function of time, and which will be referred to as the degrees of freedom moment of α.
Postulate 2.0 A nontrivial directional probabilism for the Earth’s magnetic field at a locus on a replicating strand of DNA can be derived in a directionally unstable reference frame.
Postulate 2.1 Such a directional probabilism depends on instabilities in the magnitude of the Earth’s magnetic field.
Let B⃑ᴇ be the Earth’s magnetic field in region H near the Earth’s surface, and let the position of A be bounded by this region. Then from equation (1) we observe that the position of α must also be bounded by H. Let us denote the two nucleotides of the base pair at α as α⁺ for the nucleotide on the leading strand, and α⁻ for the nucleotide on the lagging strand. And so let us denote the magnetic moment of the lagging strand nucleotide as μ⃑α⁻. Focusing on the lagging strand, let θ be the angle between μ⃑α⁻, and B⃑ᴇ, and let C⃑α be a unit vector field representing the inertial reference frame of α⁻ at any time t.
Note that in a magnetic field, and with sufficient rotational mobility, μ⃑α⁻ will rotate to align with the field as a result of orientation energy. Even in the weak magnetic field B⃑ᴇ, some amount of orientation energy will be generated, and so if the degrees of freedom at α⁻ are sufficiently high, this orientation energy will be shed by rotating μ⃑α⁻ toward alignment with B⃑ᴇ. Therefore if over some period Δt the direction of B⃑ᴇ is fixed in relation to C⃑ and α⁻ is free to rotate its position at some angular speed ω because dα⁻ ≫ 1, then the probability, P, that θ is zero at the end of this period is approximately 1.0, provided Δt is sufficiently large. We can can express this
where R₂ is a fixed rotational matrix describing the relationship between Earth’s magnetic field and the inertial reference frame of α. If, however, C⃑ is not fixed, but changes randomly with time, then the probability, P that θ is zero at the end of an equivalent period Δteq is very low, if not entirely trivial. And so we can define this probability, P0, in terms of equation (6) with the statement
However, let us consider the parameters of H. We note that the position and orientation of α within H are determined by the position and orientation of A which in turn are determined by the position and orientation of the host cell. This cell is a constituent of (or itself constitutes) a living biological organism, and by definition, H comprises the habitat of this organism. The position and orientation of an organism over time is not random, but rather is determined by distinct pattens of behavior. Therefore the position and orientation of α, which is analogous to C⃑, cannot be completely random, but must instead be some function of these patterns of behavior. It follows, then, that R₂ describes this function, and so is not random.
Consider that if the magnitude of the Earth’s magnetic field, B⃑ᴇ, is stable and nonzero, then the probability, Ps, that θf is zero at tf is greater than if B⃑ᴇ is unstable, Pu. If, however, B⃑ᴇ were to have zero magnitude, then the probability would reduce to those of random conditions. Therefore the following axiomatic statements can be made:
I. If B⃑ᴇ has zero magnitude, P {θ = 0} = P0.
II. If B⃑ᴇ has a stable magnitude, P {θ = 0} = Ps : Ps > P0.
III. If B⃑ᴇ has an unstable magnitude, P {θ = 0} = Pu : Ps > Pu > P0.
Let us then define an angle θʹα⁻ relating μ⃑α⁻ to B⃑ᴇ in terms of the third axiom:
where R₃ maps θʹ to 𝔄, an orthogonal vector subspace of R₂, described in polar coordinates² as
Note that by definition, 𝔄 is invertible and so interconversion using R₃ can go in both directions. Furthermore, note that relative to reference frame C⃑, the same that can be said of α⁻ in terms of [I], [II] and [III] can be said of a particular protein complex, χ, containing a polymerase active site, ξ, into which α will dock during replication. Therefore R₃ can also map probabilities with respect to the protein complex, and so the following holds:
Let us now consider a critical probability threshold value for θʹ that fairly characterizes the relative orientation of B⃑ᴇ:
where Pt is the threshold and θᵗ is the magnitude of θʹ there (Figure 2).
It is intuitive that exhibiting greater rotational acceleration, ωʹ, in aligning with B⃑ᴇ will increasethe probability that | θʹ | ≤ θᵗ at the end of the period Δteq. It is also intuitive that increased degrees of freedom will result in a higher ωʹ. Therefore the following two additional axiomatic statements can be made:
IV. θ ͭ is inversely related to rotational acceleration.
V. θ ͭ is inversely related to degrees of freedom.
Note that if α⁻ has both higher rotational acceleration toward B⃑ᴇ and more degrees of freedom than χ over some period Δt, it follows that θαᵗ must be less than θχᵗ .
Consider, then, the orientation energy, U⃑α⁻, of the nucleotide α⁻ over Δt, and let us simplistically model it in 𝔄 with the equation
Note then, that the torque, τ⃑α⁻, exerted on α⁻ over Δteq in 𝔄 is
and similarly
Because χ is orders of magnitude larger and more massive than α⁻, it is expected that
and furthermore that, owing to the disparity in dipole length,
and so
Furthermore, we note that after denaturation by helicase, the degrees of freedom at α⁻ are high in comparison to χ as the nucleotide is only bound to surrounding chromatin by bonds to the lagging strand nucleotides on the 3ʹ and 5ʹ sides, while the protein complex is held in position by bonds with multiple general transcription factors, and so it is expected that
Let us now consider the maximum difference between these probability threshold angles over the interval [ti, tf ], (Figure 3). Symbolizing this value θΔ, we can use this to define a more useful workspace transformation, R₄, such that for some arbitrary function of time, g
where
and where φ describes a relative difference from an effective, probabilistic value for the Earth’s magnetic field, B⃑e, with respect to C⃑ for all A ∈ H. An important consequence of [III], is that the value of θΔ depends on the degree of instability in B⃑e. Therefore characteristic fluctuations in the magnitude of B⃑e over time will significantly inform the value of φ, because θΔ establishes the scale by which momentary values of θχᵗ and θα-ͭ are contextualized.
Postulate 3.0 The internal energy at a given DNA locus changes during replication as a function of instabilities in the Earth’s magnetic field, and intrinsic properties of this locus.
Let us define t₀ as occurring at the moment α⁻ is denatured by helicase from α⁺, and so at this moment, the base pair at α is positionally fixed in the active site of the helicase protein of the replication complex. We note that as replication continues from this point, an unreplicated segment of the lagging strand gathers into a loop while the preceding unreplicated segment is read from 5ʹ to 3ʹ to generate an Okazaki fragment. Observe that as the unreplicated loop segment containing α⁻ grows, the degrees of freedom at α⁻ must increase. Once the current Okazaki fragment is extended to the preceding one, the polymerase protein moves (downstream) to the beginning of the new unreplicated segment and recommences 5ʹ to 3ʹ replication.
After the downstream repositioning of the polymerase protein, α⁻ is reeled in toward the active site ξ while the conaining loop is replicated. But let us consider a time t₁ before this reeling back in occurs, and immediately before some nontrivial change in the relative direction of B⃑e at α⁻. Additionally let t₂ be a time immediately after this nontrivial change, such that t₁ and t₂ bracket the event, and so t₁ ⪅ t₂, and dα⁻(t₁) ≈ dα⁻(t₂). Finally, let t₃ be the instant before α docks at ξ. For illustrative purposes, we will consider the case where the event at t₂ coincides with the commencement of the replication of the lagging strand unreplicated loop segment (Figure 4). Note that degrees of freedom at α⁻ must decrease between t₂ and t₃, because at the moment α⁻ docks at ξ we expect that dα⁻(t₃) ≈ dχ(t₃), and so from (25) we conclude dα⁻(t₂) > dα⁻(t₃).
Observe that work must be done to feed α⁻ into the loop, and work must also be done to reel α⁻ back in from the loop. Focusing on the effect of orientation energy we observe that as dα⁻ increases, an increasing amount of orientation energy can be shed by rotating μ⃑α⁻ toward alignment with B⃑e. This implies that, specifically with respect to orientation energy, negative work is done as t goes from t₀ to t₁ (orientation energy is shed), and subsequently positive work is done as t goes from t₂ to t₃ (orientation energy is acquired).
It is intuitive that if B⃑e were to remain directionally stable with respect to C⃑, the magnitude of the work from t₀ to t₁ (which we will hereafter call Δt₁) should equal the magnitude of the work from t₂ to t₃ (hereafter, Δt₂). However, if B⃑e is not stable, but rather changes in the moment bracketed between t₁ and t₂, we observe that
where Wᵁ with a subscript symbolizes the magnitude of the work done at α⁻ with respect to orien- tation energy.
Consider that as the lagging replication loop grows during Δt₁ and higher degrees of freedom are attained at the loci within the loop, entropy within the loop increases. Consider too that as the lagging replication loop is reeled back in toward ξ during Δt₂ and degrees of freedom are reduced at the remaining loci within the loop, entropy decreases. Let us examine the relationship between changing degrees of freedom and changing entropy by examining the change in orientation energy.
There must exist some limit to the amount of entropy any segment of the lagging strand can acquire without structural degradation, and so any quantity of entropy gained or lost by a given lagging strand segment can be viewed in terms of a proportion of this limit, which we will refer to as the orientation entropic potential. Similarly, the amount of actual change in orientation energy at α⁻ during a given period is limited by the capacity for change of orientation at α⁻, which is a function of the degrees of freedom moment across that period. Recall that by Postulate 1.0 the degrees of freedom moment of α⁻ is a function of capacity dimension, which depends on intrinsic properties of A, specific to α⁻.
Let us then use the proportion of Wᵁ to Uα⁻ to derive an expression for the orientation entropic potential:
where f is a coefficient of proportionality:
where ΔS is the change in entropy, and Sm is the orientation entropic potential. Here we can make the three following axiomatic statements:
VI. Sm depends on ΔS and intrinsic properties of α⁻, while ΔS depends on intrinsic properties of α⁻ but does not depend on Sm.
VII. f = 0 iff Wᵁ = 0, and so ΔS = 0 iff f = 0.
VIII. As Sm → 0, dα⁻ → 1.
In light of the relationships described in [VI] and [VII], and the behavior described in [VIII], we can deduce an exponential relationship between the change in entropy, and the degrees of freedom moment at α⁻:
where R₅ is a function³ describing the proportionality between the degrees of freedom moment and the exponential function of the orientation entropic potential at α⁻.
From the result of (30), we observe that there must be some net change in internal energy, ΔEα⁻ from t₀ to t₃ at α⁻, since there is a net change in work energy but no net change in entropy. We can now use (28), (34) and (36) to describe this change in energy:
Therefore let the value k be defined
such that (40) can be written more concisely
Note, however, that because we have employed φ to derive this result we are in our probability vector space, which has basis 𝔅, and so must convert back to “momentary space” for the result to have meaning. This is feasible because 𝔅 is orthogonal, and so let us define Λ (hereafter, the context energy of α) as the inverse R₄ transformation of (43):
Postulate 4.0 Changes in the context energy at a given locus during DNA replication can result in changes to tautomeric mutation rates at that locus.
Let the probability of tautomerization at α⁻ at t₀ be denoted [P {T}]₀. Observing that if tautomerization occurs at t₃, a tautomeric mutation will result, we can describe the probability of a tautomerization mutation, P {Tm}, at α⁻ at t₃ as
where EAᵗ is the activation energy required for tautomerization, and E₀ is the net energy at α⁻ discounting context energy at the moment of replication⁴. We will denote the coefficient of this proportionality as τ̇. Furthermore, we observe the rate of change of the probability
where Λʹ is the time derivative of Λ, and we will denote the constant of proportionality for this derivative as τ̈.
Defining the change in probability of tautomerization mutation as
we conclude