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Oh E, Krogmeier TJ, Schlimgen AW, Head-Marsden K. Singular Value Decomposition Quantum Algorithm for Quantum Biology. ACS PHYSICAL CHEMISTRY AU 2024; 4:393-399. [PMID: 39069975 PMCID: PMC11274286 DOI: 10.1021/acsphyschemau.4c00018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/08/2024] [Revised: 04/30/2024] [Accepted: 05/01/2024] [Indexed: 07/30/2024]
Abstract
There has been a recent interest in quantum algorithms for the modeling and prediction of nonunitary quantum dynamics using current quantum computers. The field of quantum biology is one area where these algorithms could prove to be useful as biological systems are generally intractable to treat in their complete form but amenable to an open quantum systems approach. Here, we present the application of a recently developed singular value decomposition (SVD) algorithm to two systems in quantum biology: excitonic energy transport through the Fenna-Matthews-Olson complex and the radical pair mechanism for avian navigation. We demonstrate that the SVD algorithm is capable of capturing accurate short- and long-time dynamics for these systems through implementation on a quantum simulator and conclude that while the implementation of this algorithm is beyond the reach of current quantum computers, it has the potential to be an effective tool for the future study of systems relevant to quantum biology.
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Affiliation(s)
- Emily
K. Oh
- Department of Chemistry, Washington
University in St. Louis, St. Louis, Missouri 61630, United States
| | - Timothy J. Krogmeier
- Department of Chemistry, Washington
University in St. Louis, St. Louis, Missouri 61630, United States
| | - Anthony W. Schlimgen
- Department of Chemistry, Washington
University in St. Louis, St. Louis, Missouri 61630, United States
| | - Kade Head-Marsden
- Department of Chemistry, Washington
University in St. Louis, St. Louis, Missouri 61630, United States
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2
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Cohen D. Family Constellation therapy: A nascent approach for working with non-local consciousness in a therapeutic container. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2024; 186:33-38. [PMID: 38052327 DOI: 10.1016/j.pbiomolbio.2023.11.008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2023] [Revised: 10/25/2023] [Accepted: 11/24/2023] [Indexed: 12/07/2023]
Abstract
Family Constellations are an emerging therapeutic approach for working with local and non-local consciousness. First developed by German psychoanalyst Bert Hellinger, and now practiced by thousands of licensed and un-licensed facilitators globally, Family Constellations are a transpersonal and systemically oriented therapeutic process. Their aim is to address a focus client's emotional, behavioral, relational, or somatic issues by uncovering and resolving transgenerational entanglements within their family system. The author expands on the proposal of symbiogenesis as a mediator of local and non-local consciousness to query whether applying the Observer Effect to inherited trauma may influence epigenetic marks. An expanded perspective on consciousness, life, death, and quantum fields may provide a more comprehensive framework to address therapeutic interventions for common emotional and behavioral disorders. Innovative features of Family Constellations are its phenomenological orientation, reference to family system entanglements, and potential for symptom relief through cellular mediation of ancestral memory. Family Constellations utilize techniques called representative perception and tuning-in to identify and release ancestral traumas. These are akin to remote viewing and mediumship. While the scientific basis for Family Constellations is speculative, the text references research on the quantum theory of consciousness, mediumship and remote viewing as potential supporting evidence. Four case studies are presented.
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Affiliation(s)
- Dan Cohen
- Seeing with Your Heart, http:seeingwithyourheart.com 14 Rhode Island Avenue, Providence, RI, 02906, USA.
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3
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Matarèse BFE, Rusin A, Seymour C, Mothersill C. Quantum Biology and the Potential Role of Entanglement and Tunneling in Non-Targeted Effects of Ionizing Radiation: A Review and Proposed Model. Int J Mol Sci 2023; 24:16464. [PMID: 38003655 PMCID: PMC10671017 DOI: 10.3390/ijms242216464] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2023] [Revised: 11/01/2023] [Accepted: 11/13/2023] [Indexed: 11/26/2023] Open
Abstract
It is well established that cells, tissues, and organisms exposed to low doses of ionizing radiation can induce effects in non-irradiated neighbors (non-targeted effects or NTE), but the mechanisms remain unclear. This is especially true of the initial steps leading to the release of signaling molecules contained in exosomes. Voltage-gated ion channels, photon emissions, and calcium fluxes are all involved but the precise sequence of events is not yet known. We identified what may be a quantum entanglement type of effect and this prompted us to consider whether aspects of quantum biology such as tunneling and entanglement may underlie the initial events leading to NTE. We review the field where it may be relevant to ionizing radiation processes. These include NTE, low-dose hyper-radiosensitivity, hormesis, and the adaptive response. Finally, we present a possible quantum biological-based model for NTE.
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Affiliation(s)
- Bruno F. E. Matarèse
- Department of Haematology, University of Cambridge, Cambridge CB2 1TN, UK;
- Department of Physics, University of Cambridge, Cambridge CB2 1TN, UK
| | - Andrej Rusin
- Department of Biology, McMaster University, Hamilton, ON L8S 4L8, Canada; (A.R.); (C.S.)
| | - Colin Seymour
- Department of Biology, McMaster University, Hamilton, ON L8S 4L8, Canada; (A.R.); (C.S.)
| | - Carmel Mothersill
- Department of Biology, McMaster University, Hamilton, ON L8S 4L8, Canada; (A.R.); (C.S.)
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4
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Zhang Y, Hu Z, Wang Y, Kais S. Quantum Simulation of the Radical Pair Dynamics of the Avian Compass. J Phys Chem Lett 2023; 14:832-837. [PMID: 36655839 DOI: 10.1021/acs.jpclett.2c03617] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
The simulation of open quantum dynamics on quantum circuits has attracted wide interests recently with a variety of quantum algorithms developed and demonstrated. Among these, one particular design of a unitary-dilation-based quantum algorithm is capable of simulating general and complex physical systems. In this paper, we apply this quantum algorithm to simulating the dynamics of the radical pair mechanism in the avian compass. This application is demonstrated on the IBM QASM quantum simulator. This work is the first application of any quantum algorithm to simulating the radical pair mechanism in the avian compass, which not only demonstrates the generality of the quantum algorithm but also opens new opportunities for studying the avian compass with quantum computing devices.
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Affiliation(s)
| | - Zixuan Hu
- Department of Chemistry, Department of Physics, and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana47907, United States
| | - Yuchen Wang
- Department of Chemistry, Department of Physics, and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana47907, United States
| | - Sabre Kais
- Department of Chemistry, Department of Physics, and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana47907, United States
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5
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Quantum aspects of the brain-mind relationship: A hypothesis with supporting evidence. Biosystems 2023; 223:104820. [PMID: 36442739 DOI: 10.1016/j.biosystems.2022.104820] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2022] [Revised: 11/23/2022] [Accepted: 11/24/2022] [Indexed: 11/27/2022]
Abstract
If all aspects of the mind-brain relationship were adequately explained by classical physics, then there would be no need to propose alternatives. But faced with possibly unresolvable puzzles like qualia and free will, other approaches are required. In alignment with a suggestion by Heisenberg in 1958, we propose a model whereby the world consists of two elements: Ontologically real Possibles that do not obey Aristotle's law of the excluded middle, and ontologically real Actuals that do. Based on this view, which bears resemblance to von Neumann's 1955 proposal (von Neumann, 1955), and more recently by Stapp and others (Stapp, 2007; Rosenblum and Kuttner, 2006), measurement that is registered by an observer's mind converts Possibles into Actuals. This quantum-oriented approach raises the intriguing prospect that some aspects of mind may be quantum, and that mind may play an active role in the physical world. A body of empirical evidence supports these possibilities, strengthening our proposal that the mind-brain relationship may be partially quantum.
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6
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Beshkar M. The QBIT Theory: Consciousness from Entangled Qubits. Integr Psychol Behav Sci 2022:10.1007/s12124-022-09745-w. [PMID: 36567412 DOI: 10.1007/s12124-022-09745-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 12/20/2022] [Indexed: 12/26/2022]
Abstract
How does the brain generate consciousness? The present paper is an attempt to answer this question from the perspective of the QBIT theory. In sum, the theory argues that the brain has a prior belief (P) about the stimulus that has caused a sensory representation (R) to be created in the brain. When the conditional entropy of P given R becomes less than zero, the brain becomes more than certain about (i.e. becomes conscious of) the stimulus. Conditional entropy can become negative (and thus the brain can become more than certain) only if the brain uses entangled quantum information in its computations. The QBIT theory suggests that, at the most fundamental level, consciousness is nothing but a special kind of entangled information.
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Affiliation(s)
- Majid Beshkar
- Tehran University of Medical Sciences, Tehran, Iran.
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7
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Kauffman SA, Roli A. What is consciousness? Artificial intelligence, real intelligence, quantum mind and qualia. Biol J Linn Soc Lond 2022. [DOI: 10.1093/biolinnean/blac092] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022]
Abstract
Abstract
We approach the question ‘What is consciousness?’ in a new way, not as Descartes’ ‘systematic doubt’, but as how organisms find their way in their world. Finding one’s way involves finding possible uses of features of the world that might be beneficial or avoiding those that might be harmful. ‘Possible uses of X to accomplish Y’ are ‘affordances’. The number of uses of X is indefinite (or unknown), the different uses are unordered, are not listable, and are not deducible from one another. All biological adaptations are either affordances seized by heritable variation and selection or, far faster, by the organism acting in its world finding uses of X to accomplish Y. Based on this, we reach rather astonishing conclusions:
1. Artificial general intelligence based on universal Turing machines (UTMs) is not possible, since UTMs cannot ‘find’ novel affordances.
2. Brain-mind is not purely classical physics for no classical physics system can be an analogue computer whose dynamical behaviour can be isomorphic to ‘possible uses’.
3. Brain-mind must be partly quantum—supported by increasing evidence at 6.0 to 7.3 sigma.
4. Based on Heisenberg’s interpretation of the quantum state as ‘potentia’ converted to ‘actuals’ by measurement, where this interpretation is not a substance dualism, a natural hypothesis is that mind actualizes potentia. This is supported at 5.2 sigma. Then mind’s actualizations of entangled brain-mind-world states are experienced as qualia and allow ‘seeing’ or ‘perceiving’ of uses of X to accomplish Y. We can and do jury-rig. Computers cannot.
5. Beyond familiar quantum computers, we discuss the potentialities of trans-Turing systems.
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Affiliation(s)
| | - Andrea Roli
- Department of Computer Science and Engineering, Alma Mater Studiorum Università di Bologna , Campus of Cesena, Via dell’Università, Cesena , Italy
- European Centre for Living Technology , Dorsoduro, Venezia , Italy
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8
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Zadeh-Haghighi H, Simon C. Magnetic field effects in biology from the perspective of the radical pair mechanism. J R Soc Interface 2022; 19:20220325. [PMID: 35919980 PMCID: PMC9346374 DOI: 10.1098/rsif.2022.0325] [Citation(s) in RCA: 34] [Impact Index Per Article: 17.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2022] [Accepted: 07/14/2022] [Indexed: 04/07/2023] Open
Abstract
Hundreds of studies have found that weak magnetic fields can significantly influence various biological systems. However, the underlying mechanisms behind these phenomena remain elusive. Remarkably, the magnetic energies implicated in these effects are much smaller than thermal energies. Here, we review these observations, and we suggest an explanation based on the radical pair mechanism, which involves the quantum dynamics of the electron and nuclear spins of transient radical molecules. While the radical pair mechanism has been studied in detail in the context of avian magnetoreception, the studies reviewed here show that magnetosensitivity is widespread throughout biology. We review magnetic field effects on various physiological functions, discussing static, hypomagnetic and oscillating magnetic fields, as well as isotope effects. We then review the radical pair mechanism as a potential unifying model for the described magnetic field effects, and we discuss plausible candidate molecules for the radical pairs. We review recent studies proposing that the radical pair mechanism provides explanations for isotope effects in xenon anaesthesia and lithium treatment of hyperactivity, magnetic field effects on the circadian clock, and hypomagnetic field effects on neurogenesis and microtubule assembly. We conclude by discussing future lines of investigation in this exciting new area of quantum biology.
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Affiliation(s)
- Hadi Zadeh-Haghighi
- Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada T2N 1N4
- Hotchkiss Brain Institute, University of Calgary, Calgary, Alberta, Canada T2N 1N4
| | - Christoph Simon
- Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada T2N 1N4
- Hotchkiss Brain Institute, University of Calgary, Calgary, Alberta, Canada T2N 1N4
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9
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Zadeh-Haghighi H, Simon C. Radical pairs may play a role in microtubule reorganization. Sci Rep 2022; 12:6109. [PMID: 35414166 PMCID: PMC9005667 DOI: 10.1038/s41598-022-10068-4] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2021] [Accepted: 04/01/2022] [Indexed: 12/14/2022] Open
Abstract
The exact mechanism behind general anesthesia remains an open question in neuroscience. It has been proposed that anesthetics selectively prevent consciousness and memory via acting on microtubules (MTs). It is known that the magnetic field modulates MT organization. A recent study shows that a radical pair model can explain the isotope effect in xenon-induced anesthesia and predicts magnetic field effects on anesthetic potency. Further, reactive oxygen species are also implicated in MT stability and anesthesia. Based on a simple radical pair mechanism model and a simple mathematical model of MT organization, we show that magnetic fields can modulate spin dynamics of naturally occurring radical pairs in MT. We propose that the spin dynamics influence a rate in the reaction cycle, which translates into a change in the MT density. We can reproduce magnetic field effects on the MT concentration that have been observed. Our model also predicts additional effects at slightly higher fields. Our model further predicts that the effect of zinc on the MT density exhibits isotopic dependence. The findings of this work make a connection between microtubule-based and radical pair-based quantum theories of consciousness.
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Affiliation(s)
- Hadi Zadeh-Haghighi
- Department of Physics and Astronomy, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Hotchkiss Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada.
| | - Christoph Simon
- Department of Physics and Astronomy, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Hotchkiss Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada.
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10
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Smith LD, Deviers J, Kattnig DR. Observations about utilitarian coherence in the avian compass. Sci Rep 2022; 12:6011. [PMID: 35397661 PMCID: PMC8994785 DOI: 10.1038/s41598-022-09901-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2022] [Accepted: 03/29/2022] [Indexed: 11/09/2022] Open
Abstract
It is hypothesised that the avian compass relies on spin dynamics in a recombining radical pair. Quantum coherence has been suggested as a resource to this process that nature may utilise to achieve increased compass sensitivity. To date, the true functional role of coherence in these natural systems has remained speculative, lacking insights from sufficiently complex models. Here, we investigate realistically large radical pair models with up to 21 nuclear spins, inspired by the putative magnetosensory protein cryptochrome. By varying relative radical orientations, we reveal correlations of several coherence measures with compass fidelity. Whilst electronic coherence is found to be an ineffective predictor of compass sensitivity, a robust correlation of compass sensitivity and a global coherence measure is established. The results demonstrate the importance of realistic models, and appropriate choice of coherence measure, in elucidating the quantum nature of the avian compass.
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Affiliation(s)
- Luke D Smith
- Living Systems Institute and Department of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QD, UK
| | - Jean Deviers
- Living Systems Institute and Department of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QD, UK
| | - Daniel R Kattnig
- Living Systems Institute and Department of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QD, UK.
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11
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Zadeh-Haghighi H, Simon C. Radical pairs can explain magnetic field and lithium effects on the circadian clock. Sci Rep 2022; 12:269. [PMID: 34997158 PMCID: PMC8742017 DOI: 10.1038/s41598-021-04334-0] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/23/2021] [Accepted: 12/14/2021] [Indexed: 12/21/2022] Open
Abstract
Drosophila's circadian clock can be perturbed by magnetic fields, as well as by lithium administration. Cryptochromes are critical for the circadian clock. Further, the radical pairs in cryptochrome also can explain magnetoreception in animals. Based on a simple radical pair mechanism model of the animal magnetic compass, we show that both magnetic fields and lithium can influence the spin dynamics of the naturally occurring radical pairs and hence modulate the circadian clock's rhythms. Using a simple chemical oscillator model for the circadian clock, we show that the spin dynamics influence a rate in the chemical oscillator model, which translates into a change in the circadian period. Our model can reproduce the results of two independent experiments, magnetic field and lithium effects on the circadian clock. Our model predicts that stronger magnetic fields would shorten the clock's period. We also predict that lithium influences the clock in an isotope-dependent manner. Furthermore, our model also predicts that magnetic fields and hyperfine interactions modulate oxidative stress. The findings of this work suggest that the quantum nature of radical pairs might play roles in the brain, as another piece of evidence in addition to recent results on xenon anesthesia and lithium effects on hyperactivity.
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Affiliation(s)
- Hadi Zadeh-Haghighi
- Department of Physics and Astronomy, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Hotchkiss Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada.
| | - Christoph Simon
- Department of Physics and Astronomy, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada.
- Hotchkiss Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada.
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12
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Jain R, Poonia VS, Saha K, Saha D, Ganguly S. The avian compass can be sensitive even without sustained electron spin coherence. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0778] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Theoretical studies indicating the presence of long-lived coherence in the radical pair system have engendered questions about its utilitarian role in the avian compass. In this paper, we investigate the role of electron spin coherence in a multinuclear radical pair system including its impact on compass sensitivity. We find that sustenance of long-lived electron spin coherence is unlikely in a multinuclear hyperfine environment. After probing the role of the hyperfine interactions in the compass, we affirm the hyperfine anisotropy to be an essential parameter for the necessary sensitivity required for the compass action. Thereby, we identify a parameter regime where the compass would exhibit good sensitivity even without sustained electron spin coherence.
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Affiliation(s)
- Rakshit Jain
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14850, USA
| | - Vishvendra S. Poonia
- Department of Electronics and Communication Engineering, Indian Institute of Technology Roorkee, India
| | - Kasturi Saha
- Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, India
| | - Dipankar Saha
- Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, India
| | - Swaroop Ganguly
- Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, India
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13
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Hore PJ. Upper bound on the biological effects of 50/60 Hz magnetic fields mediated by radical pairs. eLife 2019; 8:44179. [PMID: 30801245 PMCID: PMC6417859 DOI: 10.7554/elife.44179] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2018] [Accepted: 02/02/2019] [Indexed: 01/02/2023] Open
Abstract
Prolonged exposure to weak (~1 μT) extremely-low-frequency (ELF, 50/60 Hz) magnetic fields has been associated with an increased risk of childhood leukaemia. One of the few biophysical mechanisms that might account for this link involves short-lived chemical reaction intermediates known as radical pairs. In this report, we use spin dynamics simulations to derive an upper bound of 10 parts per million on the effect of a 1 μT ELF magnetic field on the yield of a radical pair reaction. By comparing this figure with the corresponding effects of changes in the strength of the Earth’s magnetic field, we conclude that if exposure to such weak 50/60 Hz magnetic fields has any effect on human biology, and results from a radical pair mechanism, then the risk should be no greater than travelling a few kilometres towards or away from the geomagnetic north or south pole.
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Affiliation(s)
- P J Hore
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford, United Kingdom
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14
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Keens RH, Bedkihal S, Kattnig DR. Magnetosensitivity in Dipolarly Coupled Three-Spin Systems. PHYSICAL REVIEW LETTERS 2018; 121:096001. [PMID: 30230901 DOI: 10.1103/physrevlett.121.096001] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2018] [Indexed: 06/08/2023]
Abstract
The radical pair mechanism is a canonical model for the magnetosensitivity of chemical reaction processes. The key ingredient of this model is the hyperfine interaction that induces a coherent mixing of singlet and triplet electron spin states in pairs of radicals, thereby facilitating magnetic field effects (MFEs) on reaction yields through spin-selective reaction channels. We show that the hyperfine interaction is not a categorical requirement to realize the sensitivity of radical reactions to weak magnetic fields. We propose that, in systems comprising three instead of two radicals, dipolar interactions provide an alternative pathway for MFEs. By considering the role of symmetries and energy level crossings, we present a model that demonstrates a directional sensitivity to fields weaker than the geomagnetic field and remarkable spikes in the reaction yield as a function of the magnetic field intensity; these effects can moreover be tuned by the exchange interaction. Our results further the current understanding of the effects of weak magnetic fields on chemical reactions, could pave the way to a clearer understanding of the mysteries of magnetoreception and other biological MFEs and motivate the design of quantum sensors. Further still, this phenomenon will affect spin systems used in quantum information processing in the solid state and may also be applicable to spintronics.
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Affiliation(s)
- Robert H Keens
- Living Systems Institute and Department of Physics, University of Exeter, Stocker Road, Exeter, Devon, EX4 4QD, United Kingdom
| | - Salil Bedkihal
- Living Systems Institute and Department of Physics, University of Exeter, Stocker Road, Exeter, Devon, EX4 4QD, United Kingdom
| | - Daniel R Kattnig
- Living Systems Institute and Department of Physics, University of Exeter, Stocker Road, Exeter, Devon, EX4 4QD, United Kingdom
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15
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Elucidation of near-resonance vibronic coherence lifetimes by nonadiabatic electronic-vibrational state character mixing. Proc Natl Acad Sci U S A 2018; 116:18263-18268. [PMID: 30093387 DOI: 10.1073/pnas.1701390115] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Abstract
Recent work suggests that the long-lived coherences observed in both natural and artificial light-harvesting systems (such as the Fenna-Matthews-Olson complex) could be attributed to the mixing of the pigments' electronic and vibrational degrees of freedom. To investigate the underlying mechanism of these long coherence lifetimes, a sophisticated description of interactions between the molecular aggregates and the nonequilibrium fluctuations in the surrounding environment is necessary. This is done by implementing the hierarchical equations of motion approach on model homodimers, a method used in the intermediate coupling regime for many molecular aggregates wherein the nonequilibrium environment phonons play nontrivial roles in exciton dynamics. Here we report a character change in the vibronic states-reflective of property mixing between the electronic and vibrational states-induced by an interplay between system coupling parameters within the exciton-vibrational near-resonance regime. This mixing dictates vital aspects of coherence lifetime; by tracking the degree of mixing, we are able to elucidate the relationship between coherence lifetime and both the electronic energy fluctuation and the vibrational relaxation dephasing pathways.
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16
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The sensitivity of a radical pair compass magnetoreceptor can be significantly amplified by radical scavengers. Sci Rep 2017; 7:11640. [PMID: 28912470 PMCID: PMC5599710 DOI: 10.1038/s41598-017-09914-7] [Citation(s) in RCA: 42] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2017] [Accepted: 07/31/2017] [Indexed: 11/08/2022] Open
Abstract
Birds have a remarkable ability to obtain navigational information from the Earth's magnetic field. The primary detection mechanism of this compass sense is uncertain but appears to involve the quantum spin dynamics of radical pairs formed transiently in cryptochrome proteins. We propose here a new version of the current model in which spin-selective recombination of the radical pair is not essential. One of the two radicals is imagined to react with a paramagnetic scavenger via spin-selective electron transfer. By means of simulations of the spin dynamics of cryptochrome-inspired radical pairs, we show that the new scheme offers two clear and important benefits. The sensitivity to a 50 μT magnetic field is greatly enhanced and, unlike the current model, the radicals can be more than 2 nm apart in the magnetoreceptor protein. The latter means that animal cryptochromes that have a tetrad (rather than a triad) of tryptophan electron donors can still be expected to be viable as magnetic compass sensors. Lifting the restriction on the rate of the spin-selective recombination reaction also means that the detrimental effects of inter-radical exchange and dipolar interactions can be minimised by placing the radicals much further apart than in the current model.
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17
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Quantifying Magnetic Sensitivity of Radical Pair Based Compass by Quantum Fisher Information. Sci Rep 2017; 7:5826. [PMID: 28725054 PMCID: PMC5517522 DOI: 10.1038/s41598-017-06187-y] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2017] [Accepted: 06/07/2017] [Indexed: 11/08/2022] Open
Abstract
The radical pair (RP) based compass is considered as one of the principal models of avian magnetoreception. Different from the conventional approach where the sensitivity of RP based compass is described by the singlet yield, we introduce the quantum Fisher information (QFI), which represents the maximum information about the magnetic field's direction extracted from the RP state, to quantify the sensitivity of RP based compass. The consistency between our results and experimental observations suggests that the QFI may serve as a measure to describe the sensitivity of RP based compass. Besides, within the framework of quantum metrology, we give two specific possible measurement schemes and find that the conventional singlet yield is corresponding to the measurement of total angular momentum. Moreover, we show that the measurement of fluctuation of the total magnetic moment is much more accurate than the singlet yield measurement, and is close to the optimal measurement scheme. Finally, the effects of entanglement and decoherence are also discussed in the spirit of our approach.
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18
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Mouloudakis K, Kominis IK. Quantum information processing in the radical-pair mechanism: Haberkorn's theory violates the Ozawa entropy bound. Phys Rev E 2017; 95:022413. [PMID: 28297997 DOI: 10.1103/physreve.95.022413] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2016] [Indexed: 11/06/2022]
Abstract
Radical-ion-pair reactions, central for understanding the avian magnetic compass and spin transport in photosynthetic reaction centers, were recently shown to be a fruitful paradigm of the new synthesis of quantum information science with biological processes. We show here that the master equation so far constituting the theoretical foundation of spin chemistry violates fundamental bounds for the entropy of quantum systems, in particular the Ozawa bound. In contrast, a recently developed theory based on quantum measurements, quantum coherence measures, and quantum retrodiction, thus exemplifying the paradigm of quantum biology, satisfies the Ozawa bound as well as the Lanford-Robinson bound on information extraction. By considering Groenewold's information, the quantum information extracted during the reaction, we reproduce the known and unravel other magnetic-field effects not conveyed by reaction yields.
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Affiliation(s)
- K Mouloudakis
- Department of Physics, University of Crete, 70013 Heraklion, Greece
| | - I K Kominis
- Department of Physics, University of Crete, 70013 Heraklion, Greece.,Institute for Theoretical and Computational Physics, University of Crete, 70013 Heraklion, Greece
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19
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Affiliation(s)
- P. J. Hore
- Department of Chemistry, University of Oxford, Oxford OX1 3QZ, United Kingdom;
| | - Henrik Mouritsen
- Institut für Biologie und Umweltwissenschaften, Carl von Ossietzky Universität Oldenburg, DE-26111 Oldenburg, Germany;
- Research Centre for Neurosensory Sciences, University of Oldenburg, DE-26111 Oldenburg, Germany
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20
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Hiscock HG, Worster S, Kattnig DR, Steers C, Jin Y, Manolopoulos DE, Mouritsen H, Hore PJ. The quantum needle of the avian magnetic compass. Proc Natl Acad Sci U S A 2016; 113:4634-9. [PMID: 27044102 PMCID: PMC4855607 DOI: 10.1073/pnas.1600341113] [Citation(s) in RCA: 101] [Impact Index Per Article: 12.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
Migratory birds have a light-dependent magnetic compass, the mechanism of which is thought to involve radical pairs formed photochemically in cryptochrome proteins in the retina. Theoretical descriptions of this compass have thus far been unable to account for the high precision with which birds are able to detect the direction of the Earth's magnetic field. Here we use coherent spin dynamics simulations to explore the behavior of realistic models of cryptochrome-based radical pairs. We show that when the spin coherence persists for longer than a few microseconds, the output of the sensor contains a sharp feature, referred to as a spike. The spike arises from avoided crossings of the quantum mechanical spin energy-levels of radicals formed in cryptochromes. Such a feature could deliver a heading precision sufficient to explain the navigational behavior of migratory birds in the wild. Our results (i) afford new insights into radical pair magnetoreception, (ii) suggest ways in which the performance of the compass could have been optimized by evolution, (iii) may provide the beginnings of an explanation for the magnetic disorientation of migratory birds exposed to anthropogenic electromagnetic noise, and (iv) suggest that radical pair magnetoreception may be more of a quantum biology phenomenon than previously realized.
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Affiliation(s)
- Hamish G Hiscock
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom
| | - Susannah Worster
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom
| | - Daniel R Kattnig
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom
| | - Charlotte Steers
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom
| | - Ye Jin
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom
| | - David E Manolopoulos
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom
| | - Henrik Mouritsen
- Institut für Biologie und Umweltwissenschaften, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany; Research Centre for Neurosensory Sciences, University of Oldenburg, 26111 Oldenburg, Germany
| | - P J Hore
- Department of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom;
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21
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Chia A, Górecka A, Kurzyński P, Paterek T, Kaszlikowski D. Coherent chemical kinetics as quantum walks. II. Radical-pair reactions in Arabidopsis thaliana. Phys Rev E 2016; 93:032408. [PMID: 27078391 DOI: 10.1103/physreve.93.032408] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2014] [Indexed: 11/07/2022]
Abstract
We apply the quantum-walk approach proposed in the preceding paper [A. Chia et al., preceding paper, Phys. Rev. E 93, 032407 (2016)] to a radical-pair reaction where realistic estimates for the intermediate transition rates are available. The well-known average hitting time from quantum walks can be adopted as a measure of how quickly the reaction occurs and we calculate this for varying degrees of dephasing in the radical pair. The time for the radical pair to react to a product is found to be independent of the amount of dephasing introduced, even in the limit of no dephasing where the transient population dynamics exhibits strong coherent oscillations. This can be seen to arise from the existence of a rate-limiting step in the reaction and we argue that in such examples, a purely classical model based on rate equations can be used for estimating the time scale of the reaction but not necessarily its population dynamics.
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Affiliation(s)
- A Chia
- Centre for Quantum Technologies, National University of Singapore, Singapore
| | - A Górecka
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
| | - P Kurzyński
- Centre for Quantum Technologies, National University of Singapore, Singapore.,Faculty of Physics, Adam Mickiewicz University in Poznań, Poznań, Poland
| | - T Paterek
- Centre for Quantum Technologies, National University of Singapore, Singapore.,Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
| | - D Kaszlikowski
- Centre for Quantum Technologies, National University of Singapore, Singapore
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22
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Xu BM, Zou J. Dark state population determines magnetic sensitivity in radical pair magnetoreception model. Sci Rep 2016; 6:22417. [PMID: 26926264 PMCID: PMC4772487 DOI: 10.1038/srep22417] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2015] [Accepted: 02/15/2016] [Indexed: 11/30/2022] Open
Abstract
What is the real role of the quantum coherence and entanglement in the radical pair (RP) compass, and what determines the singlet yield have not been fully understood. In this paper, we find that the dark states of the two-electron Zeeman energy operator (TEZE) play an important role in the RP compass. We respectively calculate the singlet yields for two initial states in this dark state basis: the coherent state and the same state just removing the dark state coherence. For the later there is neither dark state coherence nor entanglement in the whole dynamical process. Surprisingly we find that in both cases the singlet yields are the same, and based on this result, we believe that the dark state population determines the singlet yield completely, and the dark state coherence and entanglement have little contribution to it. Finally, we also find that the dark state population as well as the singlet yield anisotropy is fragile to the vertical magnetic noise. However, the orientation is robust and is even enhanced by the parallel magnetic noise because the dark states expand a decoherence-free subspace. The dark state population as well as the orientation is more robust to the hyperfine coupling noise.
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Affiliation(s)
- Bao-Ming Xu
- School of Physics, Qufu Normal University, Qufu 273165, China
| | - Jian Zou
- School of Physics, Beijing Institute of Technology, Beijing 100081, China
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23
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Molina-Espíritu M, Esquivel RO, López-Rosa S, Dehesa JS. Quantum Entanglement and Chemical Reactivity. J Chem Theory Comput 2015; 11:5144-51. [PMID: 26894237 DOI: 10.1021/acs.jctc.5b00390] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The water molecule and a hydrogenic abstraction reaction are used to explore in detail some quantum entanglement features of chemical interest. We illustrate that the energetic and quantum-information approaches are necessary for a full understanding of both the geometry of the quantum probability density of molecular systems and the evolution of a chemical reaction. The energy and entanglement hypersurfaces and contour maps of these two models show different phenomena. The energy ones reveal the well-known stable geometry of the models, whereas the entanglement ones grasp the chemical capability to transform from one state system to a new one. In the water molecule the chemical reactivity is witnessed through quantum entanglement as a local minimum indicating the bond cleavage in the dissociation process of the molecule. Finally, quantum entanglement is also useful as a chemical reactivity descriptor by detecting the transition state along the intrinsic reaction path in the hypersurface of the hydrogenic abstraction reaction corresponding to a maximally entangled state.
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24
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Hoehn RD, Nichols D, Neven H, Kais S. Neuroreceptor activation by vibration-assisted tunneling. Sci Rep 2015; 5:9990. [PMID: 25909758 PMCID: PMC4408984 DOI: 10.1038/srep09990] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2014] [Accepted: 03/20/2015] [Indexed: 11/28/2022] Open
Abstract
G protein-coupled receptors (GPCRs) constitute a large family of receptor proteins
that sense molecular signals on the exterior of a cell and activate signal
transduction pathways within the cell. Modeling how an agonist activates such a
receptor is fundamental for an understanding of a wide variety of physiological
processes and it is of tremendous value for pharmacology and drug design. Inelastic
electron tunneling spectroscopy (IETS) has been proposed as a model for the
mechanism by which olfactory GPCRs are activated by a bound agonist. We apply this
hyothesis to GPCRs within the mammalian nervous system using quantum chemical
modeling. We found that non-endogenous agonists of the serotonin receptor share a
particular IET spectral aspect both amongst each other and with the serotonin
molecule: a peak whose intensity scales with the known agonist potencies. We propose
an experiential validation of this model by utilizing lysergic acid dimethylamide
(DAM-57), an ergot derivative, and its deuterated isotopologues; we also provide
theoretical predictions for comparison to experiment. If validated our theory may
provide new avenues for guided drug design and elevate methods of in silico
potency/activity prediction.
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Affiliation(s)
- Ross D Hoehn
- Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA
| | - David Nichols
- Department of Medicinal Chemistry and Molecular Pharmacology, Purdue University, West Lafayette, IN 47907, USA
| | | | - Sabre Kais
- 1] Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA [2] Department of Physics, Purdue University, West Lafayette, IN 47907, USA [3] Department of Physics, Purdue University, West Lafayette, IN 47907, USA [4] Qatar Environment and Energy Research Institute, Qatar Foundation, Doha, Qatar
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25
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Zhang Y, Berman GP, Kais S. Sensitivity and entanglement in the avian chemical compass. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042707. [PMID: 25375523 DOI: 10.1103/physreve.90.042707] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2014] [Indexed: 06/04/2023]
Abstract
The radical pair mechanism can help to explain avian orientation and navigation. Some evidence indicates that the intensity of external magnetic fields plays an important role in avian navigation. In this paper, using a two-stage model, we demonstrate that birds could reasonably detect the directions of geomagnetic fields and gradients of these fields using a yield-based chemical compass that is sensitive enough for navigation. Also, we find that the lifetime of entanglement in this proposed compass is angle dependent and long enough to allow adequate electron transfer between molecules.
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Affiliation(s)
- Yiteng Zhang
- Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA
| | - Gennady P Berman
- Theoretical Division, LANL, and New Mexico Consortium, Los Alamos, New Mexico 87545, USA
| | - Sabre Kais
- Department of Chemistry, Department of Physics, and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA and Qatar Environment and Energy Research Institute, Qatar Foundation, Doha 5825, Qatar
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26
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Xu BM, Zou J, Li H, Li JG, Shao B. Effect of radio frequency fields on the radical pair magnetoreception model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042711. [PMID: 25375527 DOI: 10.1103/physreve.90.042711] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2014] [Indexed: 06/04/2023]
Abstract
Although the radical pair (RP) model is widely accepted for birds' orientation, the physical mechanism of it is still not fully understood. In this paper we consider the RP model in the total angular-momentum representation and clearly show a detailed mechanism for orientation. When only the vertical hyperfine (HF) coupling component is considered, analytical expressions of singlet yield angular profiles are obtained with and without considering the radio frequency field, and when the horizontal HF coupling components are considered, a numerical calculation of the singlet yield is given. Based on these analytical and numerical results we present a detailed account of the following issues: how the HF coupling induces the singlet-triplet conversion; why the vertical radio frequency field can disorient the birds, while the parallel one cannot; and why the birds are able to "train" to different field strengths. Finally, we consider a multinuclei RP model.
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Affiliation(s)
- Bao-Ming Xu
- School of Physics, Beijing Institute of Technology, Beijing 100081, China
| | - Jian Zou
- School of Physics, Beijing Institute of Technology, Beijing 100081, China
| | - Hai Li
- School of Physics, Beijing Institute of Technology, Beijing 100081, China and School of Information and Electronic Engineering, Shandong Institute of Business and Technology, Yantai 264000, China
| | - Jun-Gang Li
- School of Physics, Beijing Institute of Technology, Beijing 100081, China
| | - Bin Shao
- School of Physics, Beijing Institute of Technology, Beijing 100081, China
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27
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Kritsotakis M, Kominis IK. Retrodictive derivation of the radical-ion-pair master equation and Monte Carlo simulation with single-molecule quantum trajectories. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042719. [PMID: 25375535 DOI: 10.1103/physreve.90.042719] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2014] [Indexed: 06/04/2023]
Abstract
Radical-ion-pair reactions, central in photosynthesis and the avian magnetic compass mechanism, have been recently shown to be a paradigm system for applying quantum information science in a biochemical setting. The fundamental quantum master equation describing radical-ion-pair reactions is still under debate. Here we use quantum retrodiction to formally refine the theory put forward in the paper by Kominis [I. K. Kominis, Phys. Rev. E 83, 056118 (2011)]. We also provide a rigorous analysis of the measure of singlet-triplet coherence required for deriving the radical-pair master equation. A Monte Carlo simulation with single-molecule quantum trajectories supports the self-consistency of our approach.
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Affiliation(s)
- M Kritsotakis
- Department of Physics, University of Crete, Heraklion 71103, Greece
| | - I K Kominis
- Department of Physics, University of Crete, Heraklion 71103, Greece
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28
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Tamulis A, Grigalavicius M. Quantum entanglement in photoactive prebiotic systems. SYSTEMS AND SYNTHETIC BIOLOGY 2014; 8:117-40. [PMID: 24799958 DOI: 10.1007/s11693-014-9138-6] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2013] [Revised: 03/07/2014] [Accepted: 03/11/2014] [Indexed: 11/28/2022]
Abstract
This paper contains the review of quantum entanglement investigations in living systems, and in the quantum mechanically modelled photoactive prebiotic kernel systems. We define our modelled self-assembled supramolecular photoactive centres, composed of one or more sensitizer molecules, precursors of fatty acids and a number of water molecules, as a photoactive prebiotic kernel systems. We propose that life first emerged in the form of such minimal photoactive prebiotic kernel systems and later in the process of evolution these photoactive prebiotic kernel systems would have produced fatty acids and covered themselves with fatty acid envelopes to become the minimal cells of the Fatty Acid World. Specifically, we model self-assembling of photoactive prebiotic systems with observed quantum entanglement phenomena. We address the idea that quantum entanglement was important in the first stages of origins of life and evolution of the biospheres because simultaneously excite two prebiotic kernels in the system by appearance of two additional quantum entangled excited states, leading to faster growth and self-replication of minimal living cells. The quantum mechanically modelled possibility of synthesizing artificial self-reproducing quantum entangled prebiotic kernel systems and minimal cells also impacts the possibility of the most probable path of emergence of protocells on the Earth or elsewhere. We also examine the quantum entangled logic gates discovered in the modelled systems composed of two prebiotic kernels. Such logic gates may have application in the destruction of cancer cells or becoming building blocks of new forms of artificial cells including magnetically active ones.
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Affiliation(s)
- Arvydas Tamulis
- Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gostauto 12, Vilnius, Lithuania
| | - Mantas Grigalavicius
- Department of Radiation Biology, Institute for Cancer Research, The Norwegian Radium Hospital, Ullernchausseen 70, Oslo, Norway
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