A Quantum Model on Chemically-Physically Induced Pluripotency in Stem Cells

A quantum model on the chemically and physically induced pluripotency in stem cells is proposed. Based on the conformational Hamiltonian and the idea of slow variables (molecular torsions) slaving fast ones the conversion from the differentiate state to pluripotent state is defined as the quantum transition between conformational states. The transitional rate is calculated and an analytical form for the rate formulas is deduced. Then the dependence of the rate on the number of torsion angles of the gene and the magnitude of the rate can be estimated by comparison with protein folding. The reaction equations of the conformational change of the pluripotency genes in chemical reprogramming are given. The characteristic time of the chemical reprogramming is calculated and the result is consistent with experiments. The dependence of the transition rate on physical factors such as temperature, PH value and the volume and shape of the coherent domain is analyzed from the rate equation. It is suggested that by decreasing the coherence degree of some pluripotency genes a more effective approach to the physically induced pluripotency can be made. Introduction Induced pluripotent stem cells (iPSC) were firstly generated by Yamanaka in 2006. They isolated four key pluripotency genes Oct-3/4, SOX2, c-Myc and Klf4 essential for the production of pluripotent stem cells and successfully transformed human fibroblasts into pluripotent stem cells with a retroviral system [1]. The genomic integration of the transcription factors limits the utility of this approach because of the risk of mutations being inserted into the target cell’s genome. However, recently Deng et al reported in July 2013 that iPSC can be created chemically without any gene modification [2]. They used a cocktail of seven small-molecule compounds to induce the mouse somatic cells into stem cells (which they called Chemically iPSC or CiPSC) with a higher efficiency up to 0.2%. On the other hand, Su et al reported iPSC can be created directly through a physical approach [3][4]. They indicated that the sphere morphology helps maintaining the stemness of stem cells and proved that, due to the forced growth of cells on low attachment surface the neural progenitor cells can be generated from fibroblasts directly without introducing exogenous reprogramming factors (we call it Physically iPSC or PiPSC). The stimulus-triggered acquisition of pluripotency was proposed tentatively and retracted soon [5,6]. More rigorous experiments with larger statistics on the physical effects on stem cells and the physically-induced pluripotency are awaited for. On the other hand, although many achievements in stem cell experiments have been made, from the point of theory, the mechanism for the chemically-physically iPSC is still one of the most puzzling and confusing problems to be understand. We shall give a quantum theory regarding the acquisition of pluripotency and make an estimate on the probability of the conversion rate. The CiPSC reported in [2] will be quantitatively analyzed and calculated in more detail. Based on the discussion of the factors influencing the conversion a new model of physically-induced pluripotency will also be proposed . The fundamental processes in CiPSC include small molecules CHIR, 616452, and ESK interacting with key pluripotency-related genes Sall4 and Sox2 to enhance their expression in the early phase to activate the chemical reprogramming, and then small molecule DZNep (as a epigenetic modulator) interacting with gene Oct-4 to enhance its expression in the late phase to switch the process[2]. It is reasonable to suppose that when the small-molecule compounds are bound to the pluripotency gene it causes a sudden change in the molecular conformation (or shape) of the gene, namely a leap (quantum transition) from one of the torsional minima to the another of the molecule [7]. The same story runs in PiPSC. In the PiPSC the three-dimensional (3D) sphere formation is the only important factor to promote the reprogramming without involvement of any exogenous genes, RNAs, proteins or even small molecules. It was speculated that 3D sphere cultures may provide a microenvironment to promote cell dedifferentiation or reprogramming [3]. The further studies indicated the overexpression of gene Sox2 in 3D sphere culture plays a key role in the reprogramming event [4]. So it is natural to assume that certain conformational changes of Sox2 and other genes may have happened in the reprogramming. Therefore, both CiPSC and PiPSC can be studied on the same foot by using the theory of molecular conformational change recently proposed by us [7]. A quantum model on the induced pluripotency Model Consider the CiPSC system including a key pluripotency-related gene as a macromolecule and some small molecules interacting with it. The PiPSC can be defined as well when the small molecules are switched off. Apart from 6 translational and rotational degrees of freedom the bond lengths, bond angles, torsion (dihedral) angles of the macromolecule and the coordinate of small molecules relative to the gene and the frontier electrons form a complete set of microscopic variables to describe the system. Among these variables the torsions are slow and others are fast. Following Haken’s synergetics, the slow variables always slave the fast ones. Torsion vibration energy is 0.03-0.003 ev, the lowest in all forms of biological energies, even lower than the average thermal energy per atom at room temperature (0.04 eV in 25°C); the torsion angles are easily changed even at physiological temperature. Moreover, the torsion motion has two other important peculiarities. First, due to the strong dependence of the Shannon information quantity on oscillator frequency, the torsion vibration may play an important role in the transmission of information in the biological macromolecular system. Second, different from stretching and bending the torsion potential generally has several minima with respect to angle coordinate that correspond to several stable conformations. We have proved that the small asymmetry in potential (which does exist for a real macromolecule) would cause the strong localization of wave functions and the localized quantum conformational state can well be defined for a biological macromolecule. Therefore, the molecular conformation is defined mainly by the torsion coordinate. After conformation defined through torsion, the quantum transition between conformational states can be calculated. By the adiabatically elimination of fast variables we obtain the Hamiltonian H’ describing the conformational transition of the gene. The matrix element of H’ between quantum conformational states, namely between differentiation state k and pluripotency state ' k , is denoted by | | k H k   . Then the transition rate 2 | |   k H k can be deduced from the quantum model [7-8]. Transitional rate for induced pluripotency. The Hamiltonian of the genemolecule system can be expressed as ( , ) ( , ; ) S F H H H x x          (1) where N H is the slow variable Hamiltonian and the slow variables include the torsion angles of the pluripotency-related gene and those of the small molecule interacting with gene (the latter, the number of torsion angles for small molecules is generally small and can be neglected), F H is the fast variable Hamiltonian and the fast variables include the bond stretching / bending, the electronic variables and the coordinates of the small molecule relative to the gene. Equation (1) can be solved under the adiabatic approximation, ( , ) ( ) ( , ) M x x       (2) and these two factors satisfy ( , ; ) ( , ) ( ) ( , ) F H x x x x              (3) { ( , ) ( )} ( ) ( ) S kn kn kn H E                 (4) respectively. Here  denotes the quantum state of fast variables, and (k, n) refer to the quantum numbers of torsional conformation and torsional vibration of the gene-molecule system. Because Eq (4) is not a rigorous eigenstate of Hamiltonian HS + HF, there exist transitions between adiabatic states that result from the off–diagonal elements [9] ' ' ' ' ( ) ' | | k n S F kn kn kk nn M H H M d dx E k n H kn                     (5)