DNA Torsion-based Model of Cell Fate Phase Transitions
aa r X i v : . [ q - b i o . B M ] F e b DNA Torsion-based Model of Cell Fate Phase Transitions
Ng Shyh-Chang ∗ State Key Laboratory of Stem Cell and Reproductive Biology,Institute of Zoology, Chinese Academy of Sciences,1 Beichen West Road, Chaoyang District, Beijing 100101, China,Institute of Stem Cell and Regeneration,Chinese Academy of Sciences, Beijing, China,University of Chinese Academy of Sciences,Beijing, China
Liaofu Luo † Faculty of Physical Science and Technology,Inner Mongolia University, Hohhot 010021, China,School of Life Science and Technology,Inner Mongolia University of Science and Technology,Baotou, 014010, China (Dated: February 17, 2020)All stem cell fate transitions, including the metabolic reprogramming of stem cells and the so-matic reprogramming of fibroblasts into pluripotent stem cells, can be understood from a unifiedtheoretical model of cell fates. Each cell fate transition can be regarded as a phase transition inDNA supercoiling. However, there has been a dearth of quantitative biophysical models to explainand predict the behaviors of these phase transitions. The generalized Ising model is proposed todefine such phase transitions. The model predicts that, apart from temperature-induced phasetransitions, there exists DNA torsion frequency-induced phase transitions. Major transitions inepigenetic states, from stem cell activation to differentiation and reprogramming, can be explainedby such torsion frequency-induced phase transitions, with important implications for regenerativemedicine and medical diagnostics in the future.
I. INTRODUCTION: DNA TORSION AS THE MAIN VARIABLE IN STEM CELL FATE DECISIONSA. Stem cell fate changes
Cell biology exploded after Galileo Galilei turned his telescope inward to examine the microscopic world, and afterRobert Hooke used his microscope to observe plant and animal tissues for the first time, whereupon he describedthe existence of ‘cells’. Since then, 400 years of biology research have revealed that cells are the basic units of life,both as free-living single cells and as building blocks within complex multicellular organisms. Developmental biologyhas addressed many of the questions surrounding how single cells are organized to form multicellular tissues andorganisms. By the 21 st century, it has become clear that the fundamental principles which determine how singlecells with equivalent genomes differentiate into the cornucopia of cell-types in an organism, must lie in the governingdynamics for chromatin epigenetics, gene expression and cell fate transitions in developmental hierarchies[1].The concept that a stem cell at the top of a hierarchy can differentiate into a variety of lineages, owes muchto the pioneering ideas and experiments on haematopoietic stem cells by Till and McCulloch[2, 3]. Stem cells arecharacterized by their capacities for long-term self-renewal and multipotent differentiation. By defining the stem cellfor blood formation, they also provided an archetype for other developmental systems, including the skin, skeletalmuscle, gut, sperm and the early embryo, from which embryonic or pluripotent stem cells are derived. In adulttissues that harbor regenerative potential, stem cells will either undergo a series of fate transitions from activationto proliferation and differentiation during normal development, or switch to senescence, cell death, or canceroustransformation during aging. These fate transitions define the development and aging of every organism on Earth.Some of the most exciting experiments in developmental biology in the last century include the reprogramming ofsomatic cells. By using the techniques of somatic cell nuclear transfer (SCNT) or transgenic factor overexpression,biologists were able to reprogram a somatic cell’s differentiated fate back to that of a pluripotent stem cell[4]. Repro- ∗ [email protected] † [email protected] gramming ushered in a new era for human disease modeling and cell-based therapies. It has revolutionized the fieldof stem cell biology and regenerative medicine, and consequently our notion of the underlying plasticity in chromatin. B. Chromatin conformation state and DNA supercoiling
All eukaryotic cells package their genomes in the form of chromatin, while prokaryotic bacteria package their genomeswith similar nucleoid proteins[5]. Thus genomic DNA is highly compacted in both eukaryotic and prokaryotic cells.DNA compaction state determines its accessibility for transcription, and hence the heterogeneous gene transcriptionalstates amongst cell populations, despite possessing the same genome. Chromatin consists primarily of DNA andhistone proteins. The fundamental unit of chromatin is the nucleosome, which is made of 146 bp of DNA wrapped insupercoiled helical turns around a histone octamer. Each histone’s N-terminal tail can undergo covalent modificationswhich, in turn, control chromatin compaction, eukaryotic gene expression, and play a major role in epigenetic infor-mation transfer. For example, histone acetylation is known to locally promote open chromatin conformations andtranscription factor binding to activate local gene expression. Physiochemically, the highly basic histone N-terminaltails attractively interact with DNA to facilitate chromatin compaction. Acetylation of the histone N-terminal lysineside-chain removes a positive charge and thus weakens such electrostatic attractions, resulting in open chromatin[6, 7].Most histone modifications depend on cellular metabolism. Metabolites like acetyl-CoA, propionyl-CoA, lactyl-CoA,succinyl-CoA, ATP, ADP-ribose, S-adenosyl-methionine, etc, regulate histone acetylation, propionylation, lactylation,succinylation, phosphorylation, ADP-ribosylation, and methylation, all of which play important roles in the regulationof gene expression and chromatin conformation[8–10].Because chromatin plasticity is critical for regulating gene expression and thus cell fate transitions, chromatindynamics have been widely investigated in recent years[11, 12]. To better understand the fundamental laws that governdynamic changes in chromatin conformation, we need to develop a deeper understanding of biological macromoleculedynamics, and especially the conformational dynamics of macromolecular DNA[13–15].The DNA double helix structure is well-suited for its role as a repository of genetic information. After sequencingmost major organisms’ genomic information, a large portion of the post-genomic effort can be framed as an effortto understand how information-containing DNA is regulated to manage its information transfer to RNA[16]. Al-though the various details of histone modifications and transcription factors are absolutely important, they have alsolong overshadowed the general principle that these regulatory mechanisms all essentially revolve around regulatingchromatin conformation and hence DNA accessibility (i.e. DNA supercoiling conformation).DNA supercoiling is the most ubiquitous conformational feature of all eukaryotic and prokaryotic genomes. Thesupercoiling of DNA around histones in a left-handed direction generates about one negative Wr (writhe) pernucleosome[17]. Local DNA superhelicity not only plays a role in local chromatin conformation, but also in thestabilization of B-DNA or Z-DNA helical structures to facilitate DNA-protein interactions, especially transcription byRNA polymerases and replication by DNA polymerases[18]. DNA bond-stretching, bond-bending and torsion anglesform a complete set of microscopic variables that define DNA structures. Amongst these microscopic variables, DNAtorsion angles are the main determinants that directly affect DNA supercoiling, which affect the accessibility of DNAto polymerase or transcription factor binding[17, 18]. Thus, gene expression changes and cell fate transitions are thegeneral results of changes in DNA torsion angles.
C. DNA torsion energy and the Hamiltonian
We arrived at this conclusion based on existing biological knowledge and intuitive biochemical reasoning. Yet,to analyze the state changes in cell populations, it is imperative to introduce the theory of phase transitions andself-organization in physics. We shall begin by describing each cell as a dynamical system defined with the formalismof Hamiltonian mechanics, whereby a system is described by a set of canonical coordinates r = ( q, p ) in phase space.The time evolution of the dynamical system is uniquely defined by:d p d t = − ∂ H ∂ q , d q d t = + ∂ H ∂ p where H = H ( q , p , t ) is the Hamiltonian function, representing the total energy of the system. Hamiltonians findapplications in all areas of physics, from celestial mechanics to quantum mechanics, and especially in complex dynam-ical systems. For molecular systems, one should start from the principle of quantum mechanics and the momentum p in Hamiltonian should be replaced by an operator ∂∂ q . In this case the Hamiltonian is also an operator. With in-creasing degrees of freedom, a Hamiltonian system’s time evolution becomes more complicated and often chaotic[19].Systems with many (sometimes infinite) degrees of freedom or variables are generally hard to solve or compute exactly.Statistical mechanics methods are generally introduced to solve such many-body problems.One classic example is the Ising model[20]. The Ising model was first used to predict how ferromagnetism arisesthrough a phase transition in a system of particles, each particle with its own up or down magnetic spin. The term‘phase transition’ is most commonly used to describe abrupt transitions between different states of matter, e.g. solid,liquid, and gas. Phase transitions occur when the free energy of a system shows discontinuity with respect to somevariable, e.g. temperature or pressure. Phase transitions generally stem from the interactions of a large number ofparticles in a complex system, and does not appear in systems that are too small.Amongst the large variety of variables in the high-dimensional space of a complex dynamical system, abrupttransitions only manifest in the ‘order parameters’. An order parameter shows the degree of order across the boundariesin a phase transition system; it normally ranges between zero in one phase, and nonzero in the other phase, separatedby the critical point[20]. An example of an order parameter is the net magnetization in a ferromagnetic systemundergoing a phase transition. For liquid/gas transitions, the order parameter is the level of densities. This orderparameter concept, originally introduced in the Ginzburg–Landau theory for phase transitions in thermodynamics,was generalized by Haken to the “enslaving principle”, which states that the dynamics of fast-relaxing variables iscompletely determined by the slow-relaxing dynamics of only a few ’order parameters’[21].We shall assume DNA is the major macromolecular chain that determines a cell’s (gene expression) state. Foreach monomer or nucleotide of DNA, the bond lengths, bond angles, torsion angles { θ } , and the coordinates ofelectrons/molecules bound to the DNA, define a complete set of microscopic variables to describe its Hamiltoniansystem. Torsion vibration energy is 0 . . eV , the lowest in all forms of biological energies, even lower than theaverage thermal energy per atom at room temperature (0 . eV at 25 ◦ C ). Thus, torsion angles are easily changedeven at physiological temperature, and represent slow-relaxing or unstable variables. Following Haken’s enslavingprinciple, torsion angles would represent the ‘order parameters’ in the DNA molecular system. Moreover, the torsionmotion has two other important peculiarities. First, our earlier work had already proven that a macromolecular chain,including DNA, would manifest a rapid increase in Shannon information quantity at room temperature as its oscillatorfrequency decreases below 10 Hz , through a Bose-Einstein condensation of phonons[14]. The DNA torsion vibrationfrequency is exactly in the range below 10 Hz . Therefore, the torsion vibration conveys the largest informationquantity, as compared to bond bending and stretching, and it may play an important role in the transmission ofgenetic information and genetic noise within cells[22]. Second, unlike stretching and bending, the torsion potentialgenerally has several minima with respect to angle coordinates that correspond to several stable conformations. Inother words, a cell’s state is determined mainly by phase transitions between minima in its DNA torsion energy state,not other variables, which is a more quantifiable form of the same conclusion we arrived at with intuitive biochemicalreasoning above[14, 15].Based on this argument, we will propose a model on the mechanisms of stem cell differentiation and cell fatetransitions in general, based on phase transitions in DNA torsion. II. METHODS
The Hamiltonian of the DNA molecular system can be expressed as H = H S (cid:18) θ, ∂∂θ (cid:19) + H F (cid:18) x, ∂∂x ; θ (cid:19) (1)where H S is the slow-relaxing variable (denoted as θ ) Hamiltonian, including the torsion angles of each nucleotide, H F is the fast-relaxing variable Hamiltonian (denoted as x ) including the bond stretching / bending ,the electronicvariables, etc. The stationary Schrodinger equation H M ( θ, x ) = EM ( θ, x ) (2)can be solved under the adiabatic approximation, M ( θ, x ) = ψ ( θ ) φ ( x, θ ) (3)and these two factors satisfy H F (cid:18) x, ∂∂x ; θ (cid:19) φ α ( x, θ ) = ǫ α ( θ ) φ α ( x, θ ) (4) (cid:26) H S (cid:18) θ, ∂∂θ (cid:19) + ǫ α ( θ ) (cid:27) ψ knα ( θ ) = E knα ψ knα ( θ ) (5)respectively[13]. Here α denotes the quantum state of fast-relaxing variables, and ( k, n ) refer to the quantum numbersof torsional conformation and torsional vibration of the DNA molecular system. For a DNA molecular chain ofnucleotides, Eq(5) can be rewritten into X − ℏ I j ∂ ∂θ j + U tor ( θ , . . . , θ s ) ! ψ ( θ , . . . , θ s ) = K knα ψ knα ( θ , . . . , θ s ) (6a) U tor ( θ , θ , . . . , θ s ) = X j U ( j ) tor ( θ j ) + X j U ( j,j +1) tor ( θ j , θ j +1 ) (6b)Note that here the potential U tor ( θ , . . . , θ s ) is dependent on the fast-relaxing variable quantum number α throughthe term ǫ α ( θ ) as indicated in Eq(5). Eq(6b) shows that the torsion potential U tor ( θ , . . . , θ s ) includes two parts,the term U ( j ) tor ( θ j ) of a single nucleotide within the chain and the interaction U ( j,j +1) tor ( θ j , θ j +1 ) between neighboringnucleotides. As the interaction is switched off, the solution of Eq(6a) can be expressed as the product of eachsingle nucleotide’s wave functions, ψ knα ( θ , . . . , θ s ) = Q j ψ k j n j α j ( θ j ). The general solution of Eq(6a) is the linearcombination of ψ knα ( θ , . . . , θ s ). The quantum number k j is referred to the conformation state of the j -th nucleotideand n j -its vibration state.Based on the above formulation, we can study the DNA molecule in detail. Assume the torsion potential U ( j ) tor ( θ j )(j=1,. . . , s) has two minima V A and V B as shown in Figure 1. The corresponding vibration frequencies around twominima are denoted as ω A (in left well) and ω B B (in right well) respectively. We propose that the structural foundationof the activation/differentiation of stem cells is the existence of pairs of torsion quantum states (torsion ground-stateand torsion excited-state) for each nucleotide within a gene region. That is, we assume the quantum number kj takestwo values, k j = A or B describing these two states. Under this assumption, the macroscopic epigenetic state ofstem cells could be understood as the combinatorial result of quantum transitions between these two microscopicDNA torsion states. Of course, apart from DNA torsion, there exists other molecular variables that may influencethe activation/differentiation of stem cells. It includes chemical reactions that result in changes in protein electronicconfigurations, small molecule binding interactions, chromatin configuration and other epigenetic factors, etc. Allthese variables are either fast-relaxing variables or their influence can be ultimately represented and estimated withDNA torsion. U ( θ ) = V A + 12 Iω A ( θ − θ A ) (left) U ( θ ) = V B + 12 Iω B ( θ − θ B ) (right) III. RESULTSA. Statistical mechanics of DNA molecules
Let us assume that the system of DNA chain of nucleotides is in thermal equilibrium. We will calculate the proba-bilities of the DNA chain in two torsion states A and B . Denote the partition functions (summation of probabilities)for a single section as Z A and Z B respectively. We have[14, 23]: Z A Z B = e − β ( V A − V B ) Y A/B (cid:18) β = 1 k B T (cid:19) Θ U (Θ) ω A ω B δθ δE = V B − V A FIG. 1. Torsion potential energy U ( θ ) versus torsion angle θ . Y A/B = e β ℏ ωB − e − β ℏ ωB e β ℏ ωA − e − β ℏ ωA (7) Y A/B comes from the summation over vibration states. If the conformation vibration is neglected, then the proba-bility ratio is simply determined by V A − V B . Suppose V A < V B , then A is the favored conformation since Z A /Z B > Y A/B > ω A < ω B and Y A/B < ω A > ω B , the conformation with lower vibration frequency is more favored.When ω B is much smaller than ω A , one has Z A /Z B < B is the favored one instead of A .The above analysis was made for a single nucleotide of the DNA molecular chain. Next we will discuss thecooperativity between nucleotides in a DNA molecular chain. The partition function of the DNA molecular chain is: Z = X k l n l . . . X k s n s e − β P i E kini e − β P i U kiniki +1 ni +1 = X k = A,B X k = A,B . . . X k s = A,B e − β P i E ′ ki e − β P i U ki,ki +1 (8a)exp ( − βE ′ k i ) = exp ( − βV k i ) (cid:16) e β ℏ ω ki − e − β ℏ ω ki (cid:17) ( k i = A, B ) (8b)exp (cid:0) − β U k i k i +1 (cid:1) = h exp (cid:0) − β U k i n i k i +1 n i +1 (cid:1) i ( k i = A, B ; k i +1 = A, B ) (8c)where hi means the average over vibrational states. Here we introduce the matrix P i where h k i | P i | k i +1 i = exp ( − βE ′ k i ) exp (cid:0) − β U k i k i +1 (cid:1) Under the periodic boundary conditions one has Z = T r ( P , . . . , P s ) (9)If s sections are same, then P i = PP = (cid:18) exp ( − βE ′ A ) exp ( − β ( E ′ A + U ))exp ( − β ( E ′ B + U )) exp ( − βE ′ B ) (cid:19) ≡ (cid:18) σζ σ ζ (cid:19) exp ( − βE ′ A ) (10)( U AB = U BA = U is assumed and U AA and U BB are neglected). One has Z = T r (cid:0) P S (cid:1) = ( λ max ) S (11)where λ max is the largest eigenvalue of matrix P . The probabilities of nucleotides in state A (denoted as O A ) or B (denoted as O B ) are deduced from O A = − sβ ∂ ln Z∂E A = − β ∂ ln λ max ∂E A ′ O B = 1 − O A . (12)The calculation method given above is the same as the method used to solve the Ising model[24]. Finally we obtainthe order parameter[14, 23] O A = 12 −
12 sinh (cid:16) β ( E ′ A − E ′ B ) (cid:17)r sinh (cid:16) β ( E ′ A − E ′ B ) (cid:17) + exp( − β U ) O B = 12 + 12 sinh (cid:16) β ( E ′ A − E ′ B ) (cid:17)r sinh (cid:16) β ( E ′ A − E ′ B ) (cid:17) + exp( − β U ) (13)where E ′ A − E ′ B = V A − V B − k B T ln Y A/B . (14)Since the parameters O A or O B are decisive factors in DNA structure, they can be regarded as the order param-eters of the system. If the torsion correlation between neighboring nucleotides is strong enough, U ≫ kBT , thenexp( − β U ) = 0 O A = 1 , O B = 0 as E ′ A − E ′ B < O A = 0 , O B = 1 as E ′ A − E ′ B > A or phase B respectively). So, there exists two phases A and B given by thesymbol of E ′ A − E ′ B . Of course, the condensation may never be complete in general since the small term exp ( − β U )in Eq(13) may only approach but never equal zero.To summarize, for V A < V B (Figure 1), the system would condense into state A as the vibration is switched off.However, the vibration term Y A/B changes the result as ω A = ω b . Under | ω A − ω B ω A | ≪ k B T ln Y A/B = ℏ ω A − ω B ) ctnh ℏ ω A k B T (16)where the function ctnh x is defined by ctnh x = e x + e − x e x − e − x , an odd function decreasing with x and always larger than 1for positive x . Eqs(13) to (16) constitute our main results on the cooperative mechanism or phase transition of DNAmolecules. B. Phase transitions and applications in cell fate decisions
In statistical physics there is a theorem that states: no phase transition exists in a 1D Ising model[25]. However,from the above generalized Ising model, we have shown that a phase transition can also occur in the 1D chain, astorsion vibration is taken into account. In fact, from Eqs(13) to (16) the phase transition occurs at E ′ A = E ′ B ,namely V A − V B = ℏ ω A − ω B ) ctnh ℏ ω A k B T (17)As V A − V B < ℏ ( ω A − ω B ) ctnh ℏ ω A k B T , the chain condenses into A-phase and as V A − V B > ℏ ( ω A − ω B ) ctnh ℏ ω A k B T ,the chain condenses into B-phase.This system exhibits two kinds of phase transitions. The first is the temperature-induced phase transition (T-phasetransition), occurring at the critical temperature T c T c = ℏ ω A k B ctnh − V B − V A ) ℏ ( ω A − ω B ) (18)Since ctnh x ≥ x ), the phase transition only exists under the condition V B − V A > ℏ ( ω A − ωB ) > V A − V B > ℏ ( ω A − ω B ) >
0. For example, as V B > V A the chain is condensed in A-phase as T < T C T and inB-phase as
T > T c for ω frequencies that satisfy the above conditions (Figure 2). The prediction that there existsa temperature-induced phase-transition provides an experimental checkpoint for our present theory. Moreover, onemay deduce the ratio V B − V A ) ℏ ( ω A − ω B ) from Eq(18) by using the measured value of the critical transition temperature T c .The second type of phase transition predicted for the DNA molecular chain is the torsion–induced phase transition.One can adjust the frequency ω A or ω B of torsion potential (and/or V B − V A ) to obtain the phase transition. Forexample, as V A < V B , the chain is condensed in state A as ω A = ω B . However, we can adjust ω − B to induce an ω -phase transition. Suppose ℏ ω A k B = 0 . ω -phase transition from state A to B can be realizedthrough decreasing ω B by δω = ℏ (0 . − . V B − V A ). In fact, by setting δω = ω A − ω B , the critical torsion point( δω ) c at a given temperature T is defined by ( δω ) c = 2( V B − V A ) ℏ ctnh ℏ ω A k B T . (19)which is deduced from Eq(17) (Figure 2). The prediction of ω -phase-transitions provides another experimental check-point for the present theory.To examine the experimentally verifiable and quantitative relationship between the critical temperature Tc and thecritical torsion point in detail, it would be useful to further simplify Eq(17):Let ℏ ( ω A − ω B )2( V B − V A ) = ∆ ω (dimensionless torsion ratio) ℏ ω A k B = α (constant)then ∆ ω = tanh aT . (20)This is a sigmoidal function that would plateau out at extreme values of T. But we should note that for torsionvibration energies of 0 . . ev , the constant a is estimated to be ∼ K . If we plot the possible distributionof values of the critical torsion ratio ∆ ω for T in the physiological range of 273 to 323 K (0 to 50 ◦ C ), we obtain adecreasing curve:Moreover we note that d∆ ω d T = − αT sech αT .Suppose there exists a molecule in the microenvironment that can induce DNA torsion changes, and its relationshipwith torsion can be represented linearly as∆ ω = ∆ o + bC where ∆ o is the initial torsion ratio, C is the concentration of the molecule, and b is the linearcoefficientthen d C d T = − αbT sech αT . (21) Temperature T ( K ) T o r s i o n r a t i o ∆ ω
270 280 290 300 310 320 3300 . . . . . . . .
36 Phase A Phase B ∆ ω = tanh (cid:0) αT (cid:1) , a = 100 K FIG. 2. Phase diagram for any DNA region, with the order parameters defined as temperature T and the torsion ratio ∆ ω . Basedon Eq(20), a decreasing curve separates phases A and B , indicating that as the temperature T increases, the critical torsionratio ∆ ω required for DNA to undergo a phase transition becomes smaller. The torsion ratio is defined by ∆ ω = ℏ ( ω A − ω B )2( V B − V A ) ,where ℏ is the reduced Planck constant, ( ω A − ω B ) is the change in torsion frequency, and ( V B − V A ) is the difference in torsionpotential energy. With Eq(20) and Eq(21), one should be able to experimentally verify if the critical molecular concentration C vs.critical temperature T curve follows the quantitative relationship predicted by this theory.Suppose we define the phase index ( P I ) as PI = 1 − ℏ ( ω B − ω A )2( V A − V B ) ctnh ℏ ω A k B T . (22)In the vicinity of the critical point, by inserting Eq(18) into (19) we obtain PI = 1 − ctnh ℏ ω A k B T ctnh ℏ ω A k B T C . (23) P I is therefore an observable parameter that indicates where in phase space the system resides, relative to thecritical phase transition point. When T = T C ( P I = 0), the system is at the critical phase transition point. When
T < T C ( P I < A . When T < T C ( P I > B .Abrupt cell fate changes during somatic reprogramming, directed differentiation or activation of stem cells (alltypically cultured at ∼ ◦ C ), provide a variety of vignettes that suggest the existence of cellular phase transitions[25].In fact, there are many molecules that can induce DNA torsion changes to trigger ω -phase transitions. In particular themetabolic reprogramming of stem cells, including pluripotent stem cells and muscle stem cells, suggest that non-lineage-specific factors could trigger such ω -phase transitions. For example, histones are modified by histone acetyltransferases(HAT) and deacetylases (HDAC) to direct chromatin conformation and regulate local supercoiling of DNA[6, 7]. Here,the metabolite-based regulation of HATs and HDACs by acetyl-CoA, NAD+, and short-chain fatty acids can directlyinduce changes in DNA torsion potential to trigger ω -phase transitions in cell fate[26]. Histone methylation andits regulation by histone methyltransferases (HMT) and JmjC-domain-containing histone demethylases (JHDM) arealternative mechanisms to regulate chromatin conformation and local DNA supercoiling. Similar to histone acetylation,metabolite-based regulation of HMTs and JHDMs via S-adenosyl-methionine, α -ketoglutarate, ascorbate and F e can also induce changes in DNA torsion potential to trigger cell fate transitions[27]. It is also well-known that directgenetic mutation of histones or histone modifying enzymes to alter the chromatin conformation and DNA supercoilingcan induce drastic changes in stem cell fate decisions and organismal development[28]. Some anthracycline drugs suchas doxorubicin or idarubicin can directly intercalate between nucleotide base pairs in DNA and alter DNA torsionpotential to induce mitotic arrest or differentiation in cancer cells.[12]From a general theoretical point of view, all molecule-induced changes in the torsion parameters ω A , ω B , V A , V B cancause a phase transition. These changes can occur through the change of torsion potential U tor ( θ , . . . , θ S ). In fact,all the metabolite- or genetic- or drug-induced torsion changes discussed above could be ascribed to the ǫ α ( θ ) termfor electron motion (between histones and DNA), and this term was already included as a part of the potential term U tor ( θ . . . , θ S ).NB: Molecular binding is an important factor in stem cell activation and differentiation. The binding or unbindingof (small) molecules to DNA creates an additional term in the Hamiltonian Eqs(4) and (5). The partition function ofthe DNA molecular chain will be changed from Eq(24a) to Z = Tr ( P P ′ j . . . , P S ) (24a) P j ′ = P j ⊗ (cid:18) e − βV c e − βV d (cid:19) == (cid:18) exp( − βE ′ A ) exp ( − β ( E ′ A + U ))exp ( − β ( E ′ B + U )) exp( − βE ′ B ) (cid:19) ⊗ (cid:18) e − βV c e − βV d (cid:19) (24b)where ⊗ means outer product, V c and V d are the molecular energies in binding and unbinding state respectively.From Eq(24a), one has Z = ( λ max ) s (cid:0) e − βV c + e − βV d (cid:1) (25)instead of Eq(24a). However, the order parameter equations Eq(25) remain unchanged. Therefore, the proposedtheory can also be used in analyzing cell fate transitions initiated by small molecules. IV. DISCUSSION
Thus all mechanisms of cell fate transitions, be it the reprogramming of fibroblasts into iPSCs, or the activationof stem cells into proliferative progenitors, or the differentiation of stem cells into terminally differentiated cells, orthe transdifferentiation of cells between different lineages, could be ascribed to quantum transitions between differentDNA macromolecular torsion states. This paradigm could also quantifiably explain why, besides lineage-specificsignaling factors and transcription factors, some apparently non-specific factors like metabolites, histones and DNAintercalating drugs can also trigger abrupt changes in cell fate programming. A general corollary of this new paradigmis that a full understanding of all the physiochemical and metabolic factors that control DNA macromolecular torsioncould help us improve our speed and efficiency in directing cell fates for molecular medicine.One might notice that while we have only discussed two phases of local DNA torsion, A and B , there exists alarge number of possible cell fates. This is largely because chromosomal DNA is actually not a homogeneous chain assimply assumed in this model, but peppered with heterogeneous elements such as insulators, tandem repeat elements(constitutive heterochromatin), and super-enhancer regions between the gene cluster regions. Hence each insulator,tandem repeat element, enhancer or gene cluster region could undergo local A-B phase transitions with its own uniquetorsion conditions and critical torsion points. This would produce a huge combinatorial diversity of DNA regions ineither phase A or B , thus generating a large number of possible cell fates. Others have also noted that tissuedevelopment can be represented as a series of bifurcations into two possibilities on a trajectory in the Waddington“epigenetic” landscape[25], which is consistent with our two-phase simplification of the DNA torsion theory.Recent advances in FTIR imaging, which is a label-free technique being used for detecting specific chemical compo-sitions via their bonds’ vibrational energy spectra and which can now attain nanometer resolutions, should permit usto repurpose the FTIR technique to observe the DNA torsion energy profile in local regions of a cell’s chromosomes[29].If our theory is true, we should then be able to non-invasively observe and predict a cell’s epigenetic state and fatetransitions, simply based on just its DNA torsion energy profile. Combined with the latest advances in partial wavespectroscopic (PWS) microscopy to map local chromatin density profiles[30], this could have important implicationsfor medical diagnostics in the future.We should also be able to induce abrupt cellular fate transitions with critical concentrations of drugs that specificallyalter the DNA torsional state, either by directly intercalating with DNA and altering its supercoiling, or by indirectly0regulating the histone modifications to alter the supercoiling of DNA around nucleosomes. This could have importantimplications for stem cell therapies in regenerative medicine, anti-aging therapies in gerontology, cancer therapiesin oncology, or any therapeutics that involve cellular plasticity. In fact, the DNA intercalating doxorubicin andother anthracyclines are already well-established anti-cancer drugs, with potent effects on the chromatin state andDNA supercoiling[12]. Previous studies indicate that they can treat leukemia, not simply by inducing apoptosis aspreviously thought, but by inducing leukemic stem cell differentiation[31]. Chromatin-related cancer drugs that inhibitthe HMT/JHDM/HAT/HDAC enzymes, some of which have passed clinical trials, could work in a similar fashion.Recent findings also suggest that the transition from a proliferative state to aging-associated senescence is due toreversible defects in chromatin maintenance, with implications for the potential reversal and treatment of aging andprogeria syndromes[32]. Even bacterial DNA supercoiling has an impact on bacterial growth and dormancy[33], withobvious implications for our ongoing war with infectious epidemics, and our constant search for new antibiotics andnew therapeutic windows to target multi-drug resistant bacteria. New methods to control cell fate via DNA torsiontransitions could represent a new class of strategies for medical therapeutics in the future.With these diagnostic and therapeutic applications in mind, future work could focus on molecular simulations andprecise measurements of DNA torsion potential minima. Such efforts could be based on local chromatin torsion anddensity profiles and their associated metabolic (or other physiochemical) variables, in combination with the genomicinformation networks they encode, to accurately predict any cellular state and cell fate transition. Given the generalityof DNA supercoiling and cell fate transitions, we expect our DNA torsion-based cellular phase transition theory tobe relevant to almost every field of medicine. [1] G. Q. Daley, Stem cells and the evolving notion of cellular identity, Philosophical Transactions of the Royal Society B: Biological Sciences , 20140376 (2015).[2] E. A. McCulloch and J. E. Till, The radiation sensitivity of normal mouse bone marrow cells, determined by quantitativemarrow transplantation into irradiated mice, Radiation Research , 115 (1960), https://doi.org/10.2307/3570877.[3] J. E. Till and E. A. McCulloch, A direct measurement of the radiation sensitivity of normal mouse bone marrow cells,Radiation Research , 213 (1961).[4] D. A. Robinton and G. Q. Daley, The promise of induced pluripotent stem cells in research and therapy,Nature , 295 (2012).[5] A. Gaspar-Maia, A. Alajem, E. Meshorer, and M. Ramalho-Santos, Open chromatin in pluripotency and reprogramming,Nature Reviews Molecular Cell Biology , 36 (2010).[6] M. Grunstein, Histone acetylation in chromatin structure and transcription, Nature , 349 (1997).[7] W. L. Cheung, S. D. Briggs, and C. D. Allis, Acetylation and chromosomal functions,Current Opinion in Cell Biology , 326 (2000).[8] N. Shyh-Chang, G. Q. Daley, and L. C. Cantley, Stem cell metabolism in tissue development and aging,Development , 2535 (2013).[9] J. G. Ryall, T. Cliff, S. Dalton, and V. Sartorelli, Metabolic reprogramming of stem cell epigenetics,Cell Stem Cell , 651 (2015).[10] H. Zhang, K. J. Menzies, and J. Auwerx, The role of mitochondria in stem cell fate and aging, Development ,10.1242/dev.143420 (2018), https://dev.biologists.org/content/145/8/dev143420.full.pdf.[11] X. Liu, M. Li, X. Xia, X. Li, and Z. Chen, Mechanism of chromatin remodelling revealed by the Snf2-nucleosome structure.,Nature , 440 (2017).[12] F. Yang, S. S. Teves, C. J. Kemp, and S. Henikoff, Doxorubicin, DNA torsion, and chromatin dynamics,Biochimica et Biophysica Acta (BBA) - Reviews on Cancer , 84 (2014).[13] L. Luo, Quantum theory on protein folding, Science China Physics, Mechanics and Astronomy , 458 (2014).[14] L. Luo, Conformation dynamics of macromolecules, International Journal of Quantum Chemistry , 435 (1987),https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.560320404.[15] L. Luo and J. Lv, Quantum conformational transition in biological macromolecule, Quantitative Biology , 143 (2016).[16] K. A. Frazer, Decoding the human genome, Genome Research , 1599 (2012).[17] T. Nikitina, D. Norouzi, S. A. Grigoryev, and V. B. Zhurkin, DNA topology in chromatin is defined by nucleosome spacing,Science Advances , e1700957 (2017).[18] S. Ravichandran, V. K. Subramani, and K. K. Kim, Z-DNA in the genome: from structure to disease,Biophysical Reviews , 383 (2019).[19] V. Latora, A. Rapisarda, and S. Ruffo, Superdiffusion and out-of-equilibrium chaotic dynamics with many degrees offreedoms, Phys. Rev. Lett. , 2104 (1999).[20] S. Torquato, Toward an ising model of cancer and beyond, Physical Biology , 015017 (2011).[21] H. Haken, Synergetics, in Self-Organizing Systems , edited by F. E. Yates, A. Garfinkel, D. O. Walter, and G. B. Yates(Springer US, 1987) pp. 417–434.[22] G. Tkaˇcik, C. G. Callan, and W. Bialek, Information capacity of genetic regulatory elements, Physical Review E ,10.1103/physreve.78.011910 (2008).[23] L. Luo, C.-H. Luan, K. Ma, and J. Wang, A theory of conformation electron field, noop Acta Biophysica Sinica , 170 (1985).[24] K. Huang, Statistical Mechanics, 2nd Edition (John Wiley & Sons, 1987).[25] L. Xu, K. Zhang, and J. Wang, Exploring the mechanisms of differentiation, dedifferentiation, reprogramming and transd-ifferentiation, PLoS ONE , e105216 (2014).[26] V. Morales and H. Richard-Foy, Role of histone n-terminal tails and their acetylation in nucleosome dynamics,Molecular and Cellular Biology , 7230 (2000).[27] J. Brumbaugh, I. S. Kim, F. Ji, A. J. Huebner, B. D. Stefano, B. A. Schwarz, J. Charlton, A. Coffey, J. Choi, R. M. Walsh,J. W. Schindler, A. Anselmo, A. Meissner, R. I. Sadreyev, B. E. Bernstein, H. Hock, and K. Hochedlinger, Inducible histonek-to-m mutations are dynamic tools to probe the physiological role of site-specific histone methylation in vitro and in vivo,Nature Cell Biology , 1449 (2019).[28] M. Buschbeck and S. B. Hake, Variants of core histones and their roles in cell fate decisions, development and cancer,Nature Reviews Molecular Cell Biology , 299 (2017).[29] G. C. Ajaezi, M. Eisele, F. Contu, S. Lal, A. Rangel-Pozzo, S. Mai, and K. M. Gough, Near-field infrared nanospec-troscopy and super-resolution fluorescence microscopy enable complementary nanoscale analyses of lymphocyte nuclei,Analyst , 5926 (2018).[30] S. Gladstein, A. Stawarz, L. Almassalha, L. Cherkezyan, J. Chandler, X. Zhou, H. Subramanian, and V. Backman,Measuring nanoscale chromatin heterogeneity with partial wave spectroscopic microscopy, in Methods in Molecular Biology ,Methods in Molecular Biology, Vol. 1745 (Humana Press Inc, 2018) pp. 337–360.[31] A. K. Larsen, Involvement of DNA topoisomerases and DNA topoisomerase inhibitors in the induction of leukemia celldifferentiation, Annals of Oncology , 679 (1994).[32] T. Chandra and K. Kirschner, Chromosome organisation during ageing and senescence,Current Opinion in Cell Biology , 161 (2016).[33] A. Conter, Plasmid DNA supercoiling and survival in long-term cultures of escherichia coli: Role of NaCl,Journal of Bacteriology185