Approaching Magnetic Field Effects in Biology using the Radical Pair Mechanism Jeffrey M. Canfield, Ph.D. Department of Physics University of Illinois at Urbana-Champaign, 1997 Prof.P.G.Debrunner, Advisor Prof.R.L.Belford, Co-Advisor UMI (Dissertation Abstracts) control # AAT9812544 (vol.59, no.9, 1999) call 1-800-521-3042 from US or 1-800-343-5299 from Canada or see http://www.umi.com/hp/Support/DServices/order/ to order also available at many university libraries (OCLC/WorldCat lists some) Table of Contents Chapter 1 Introduction.................................................................1 1.1 Biological Magnetic Field Effects........................................4 1.2 Biological Radical Systems...............................................5 1.3 Earth's Magnetic Field..................................................10 2 Radical Pair Mechanism......................................................13 2.1 Schrodinger Formalism...................................................14 2.1.1 Cage Dynamics.....................................................15 2.2 Liouville Formalism.....................................................16 2.3 Sample Results for Simple System........................................18 2.4 Relationships between Formalisms........................................20 2.5 Oscillating Field Treatments............................................23 2.5.1 Numerical Integration.............................................23 2.5.2 Rotating Frame....................................................24 2.5.3 Perturbation Methods..............................................25 3 Schrodinger Perturbation Method.............................................27 3.1 General Approach........................................................27 3.1.1 Derivation........................................................27 3.1.2 Discussion........................................................29 3.2 Explicit Evaluation to Second Order.....................................31 3.2.1 Derivation........................................................31 3.2.2 Discussion........................................................32 3.2.3 Selection Rules...................................................35 3.3 Sample Results for Simple System........................................37 3.3.1 Comparison of Methods.............................................37 3.3.2 Execution Times...................................................38 3.3.3 Other Results.....................................................39 3.4 Conclusion..............................................................41 4 Liouville Perturbation Method...............................................50 4.1 Derivation..............................................................50 4.2 Summary of Results......................................................52 4.3 Implementation Details..................................................52 4.4 Execution Times.........................................................53 4.5 Comparison of Methods...................................................54 4.6 Conclusion..............................................................56 5 Mixed Perturbation Methods..................................................60 5.1 VD Method...............................................................61 5.1.1 Derivation........................................................62 5.1.2 Discussion........................................................67 5.1.3 Calculating Complex Eigenvalues En................................68 5.1.4 Implications......................................................69 5.2 VK Method...............................................................70 5.2.1 Derivation........................................................70 5.3 Comparison of Methods...................................................74 6 B12 Systems.................................................................79 6.1 Spin Hamiltonians.......................................................80 6.2 Energy Level Diagrams...................................................84 6.2.1 High J Limit......................................................84 6.2.2 Isotropic case....................................................84 6.2.3 z case............................................................85 6.2.4 Corrections to z case levels......................................87 6.2.4.1 Nitrogen Splittings.......................................87 6.2.4.2 Bz Splittings.............................................88 6.2.4.3 Other Splittings..........................................88 6.2.5 2x2 Systems.......................................................89 6.2.6 Finding Effective 2x2 Systems.....................................90 6.3 Yields..................................................................92 6.3.1 Yields for the z case.............................................93 6.3.2 Corrections to z case yields due to level repulsions..............95 6.4 Energy Levels Near J=326 MHz............................................97 6.4.1 Level Repulsions..................................................97 6.4.2 Energy Levels at Jmin when Bx != 0...............................100 6.4.2.1 Axial Case...............................................100 6.4.2.2 Rhombic Case.............................................101 6.5 Effects on Yields Near J=326 MHz.......................................102 6.5.1 Zero-Field Effects...............................................102 6.5.2 Orientation-Dependent Field Effects..............................103 6.5.2.1 Axial Case...............................................103 6.5.2.2 Rhombic Case.............................................105 6.5.3 Oscillating-Field Effects........................................107 6.5.4 Powder Averaged Field Effects....................................110 6.6 Powder Averaged Effects for General J..................................111 6.7 Conclusion.............................................................113 7 Conclusion.................................................................124 Appendix A Relationships among Yields.................................................126 A.1 Steady-State Yields....................................................126 A.2 Time-Dependent Yields..................................................127 A.2.1 rho(t) is Hermitian..............................................127 A.2.2 Phi_ST(t)=3 Phi_TS(t) when k=k_S=k_T.............................128 A.2.3 k_S Phi_ST(t)=3 k_T Phi_TS(t) when k_S != k_T....................129 B Rotating Frame in the Liouville Formalism..................................132 C Relationships among the Lineshape Functions A-D............................135 D Magnitudes of Extrema in the Schrodinger or Mixed Methods..................137 E Time-Dependent Yields in the Schrodinger or Mixed Methods..................140 F Phase/Orientation Averaging in the Schrodinger Method......................143 F.1 Phase Averaging........................................................144 F.2 Orientation Averaging..................................................144 G Relaxation in the Liouville Formalism......................................147 G.1 Redfield Matrices......................................................147 G.2 Relaxation at First Glance.............................................148 G.3 Low Field T1=T2 Relaxation.............................................149 G.4 High Field T1 != T2 Relaxation.........................................151 G.5 Using Relaxation in the Liouville Perturbation Method..................152 H Repulsion-Induced Corrections to Yields Revisited..........................154 H.1 k_S != k_T Effects.....................................................156 H.2 Linewidths Revisited...................................................158 I Simple System in the Rotating Frame........................................159 I.1 Steady Field Effects...................................................160 I.2 Oscillating Field Effects..............................................161 I.3 Estimating Level Repulsions............................................162 I.4 Oscillating Field Effects Revisited....................................164 J Oscillating Field Effect Perturbation Series Examples......................169 J.1 Preliminaries..........................................................169 J.2 Effects in Simple System...............................................171 J.2.1 First Order Terms................................................173 J.2.2 Second Order Terms...............................................175 J.2.3 Comments.........................................................176 J.3 Effects in B12 System..................................................177 J.3.1 First Order Terms................................................178 J.3.2 Second Order Terms...............................................180 J.3.3 Comments.........................................................181 K EPR Simulations............................................................184 K.1 Direct Diagonalization.................................................185 K.2 Perturbation Method....................................................186 K.2.1 Diagonalizing Zeeman Terms.......................................189 K.3 Refinements............................................................192 K.3.1 Multiple Linewidths..............................................192 K.3.2 Linewidth Anisotropy.............................................193 K.4 Sample Spectra.........................................................195 K.5 Fermi's Golden Rule....................................................195 K.5.1 A Method for Finding Interesting Oscillating Field Effects.......198 References...................................................................203 Curriculum Vitae.............................................................212 Chapters 1, 2, 5, and 7 and the Appendices contain previously unpublished work. Some excerpts of the thesis appear below. Abstract The goal of my graduate work has been to try to understand or explain some of the reported magnetic field effects in biology (see Chapter 1 for examples) using the radical pair mechanism, a quantum mechanical mechanism known for over 20 years that lets the yields of certain radical pair reactions depend on the applied magnetic field [1,2,3]. This goal seems reasonable considering the known roles of many biological free radicals in cancer, disease, aging, development, and cellular signaling, the constant reminders in the media to take anti-oxidant vitamins to protect against dangerous free radicals, and the success of the radical pair mechanism in explaining magnetic field effects in photosynthetic reaction centers. The radical pair mechanism (as detailed in Chapter 2) occurs when a pair of radicals forms a cage radical pair, a system composed of two unpaired electron spins whose spin motion is affected by nearby nuclear spins via the hyperfine interaction, by applied magnetic fields via the Zeeman effect, and by each other via the exchange and dipole interactions. The spin motion varies in singlet/triplet character, and if the chemical reaction is more (or less) favorable in a singlet/triplet state, the reaction rate can depend on applied magnetic fields, even ones very weak and near earth-strength. To approach the above goal, during my graduate work I have developed and published several new perturbation treatments for combinations of steady and oscillating magnetic fields in the radical pair mechanism: one based on the Schrodinger Equation (see [4,5] or Chapter 3), another based on the Liouville Equation (see [6] or Chapter 4), and two more mixed perturbation methods that bridge the gap between the Schrodinger and Liouville formalisms (see Chapter 5). All of these iterative approaches can be used to calculate singlet-to-triplet yields when the strength of the oscillating magnetic field is weak compared to the other terms in the spin Hamiltonian. This range occurs both in the natural magnetic environment (where oscillating fields tend to be smaller than 0.03 G and steady fields tend to be near 0.5 G) and in many man-made environments. Thus, these perturbation treatments should be applicable both in studies of magnetic sensory mechanisms in animals and in studies of health effects of electromagnetic fields. These perturbation methods also allow much faster calculation of singlet-to-triplet yields than do numerical integration methods and are more generally applicable than the rotating frame treatment, allowing treatment of anisotropic spin Hamiltonians and treatment of multiple oscillating fields at any orientation with respect to the steady field. Finally, the different perturbation methods complement each other; that is, the Liouville Equation method yields a more efficient and reliable computer algorithm while the Schrodinger Equation method yields more insight into how effects of steady and oscillating magnetic fields occur and can more easily be used to generate analytical expressions for field and frequency dependences of singlet-to-triplet yields. Also, while the Liouville formalism allows one to calculate effects on steady and oscillating field sensitivity of different escape rates (k_S and k_T) for singlet and triplet pairs and can be generalized to include relaxation, the Schrodinger formalism lets one treat both exponential and Noyes time-dependences. Together these new methods can be quite useful for studying biological effects of oscillating magnetic fields, and their sample results show a number of behaviors in the singlet-to-triplet yields (such as saturation effects, oscillating field strength and frequency resonances, and steady field strength- and orientation- dependent frequency shifts) that, while typical in quantum mechanics or magnetic resonance, seem surprising in biology and may account for conflicts in the magnetic field bioeffects literature. Also during my graduate work I have used EPR (Electron Paramagnetic Resonance) data for the cage radical pairs formed by the homolytic cleavage of Co-C bonds in several coenzyme B12 dependent enzymes to calculate effects of earth-strength steady and oscillating magnetic fields on their singlet-to-triplet yields via the radical pair mechanism (see [7] or Chapter 6). Energy level repulsions and the state-mixing they induce are found to be very important for determining overall sizes of effects and lower bounds on oscillating-field frequencies that can cause effects in such systems. B12 and similar systems with nearly axial zero-field spin Hamiltonians, dominated by terms over 100 times larger than Zeeman terms due to earth-strength steady fields, if under relatively immobile conditions of long lifetime, slow molecular tumbling, and slow spin relaxation, may be useful as biological sensors of both steady and oscillating fields that occur in nature, since the yields calculated show sensitivity to steady field strength (even after powder averaging) and orientation, and undergo steady-field-dependent shifts in the oscillating-field frequencies of maximal effect. Thus, since B12 is used by a number of enzymes (including ribonucleotide reductase, which converts RNA to DNA nucleotides; methyl malonyl CoA mutase, which controls the metabolism of certain fatty acids in mammals; and methionine synthase, which in mammals is used to regenerate active methyl groups on S-adenosyl methionine, which is involved in DNA methylation, melatonin and epinephrine synthesis, myelination, and methylation of chemotaxis proteins) and since some of these B12-dependent processes have been reported to be influenced by magnetic fields, coenzyme B12 may be an interesting candidate target for magnetic field effects in biology. Chapter 1 - Introduction This work was stimulated in part by the continued interest in the magnetic sense in animals [8,9] as well as the recent concern over health effects of nonionizing oscillating electromagnetic fields [10,11]. The magnetic sense of animals has been reported for some time, and the body of literature describing magnetic field effects on animal behavior and orientation is enormous (see, for example, [12,13,14,15,16,17], and the references therein). In some animals the mechanism is fairly well understood; for example, magnetotactic bacteria have deposits of magnetite that simply align with the ambient field as would a compass needle, and sharks and rays detect the tiny electric fields induced by their motion through magnetic fields [13]. In other animals such as amphibians, reptiles, birds, and mammals, however, the mechanism is not well understood, many effects have not been reproduced, and there is debate on whether effects can occur at all. Also, in recent years there has been a growing controversy over whether or not magnetic and electromagnetic fields from power lines, electric blankets, computers, etc. have adverse effects, such as cancer or birth defects, on humans. Although effects at large oscillating magnetic fields are often thermally induced [18, pp.18,152], this thesis tries to explain effects of weak magnetic fields from a biophysical standpoint using the radical pair mechanism, a physical mechanism that has been proposed to account for biological effects of magnetic fields [19,20,21,22,23,24,25,26,27]. This mechanism was first recognized in the late sixties, allows weak magnetic fields to affect the rates of certain chemical reactions [1,3], and has proven quite successful in explaining the effects of steady as well as microwave magnetic fields on the photosynthetic reaction center (see, for example, [28,29,30,31] and [2, pp.96-100]) when the electron transfer chain is blocked by reducing the iron-ubiquinone primary acceptor complex. Also, even though many radical pair processes occur in photochemistry [2, pp.74-84], photostimulation is not a prerequisite for this mechanism. The radical pair mechanism occurs when two free radicals (two molecules each with just one unpaired electron spin) are near enough that (i) there is a high probability they will encounter and react with each other rather than with different free radicals and (ii) their spin dynamics is dominated by the interaction of their spins with each other, nearby nuclear spins, and the external magnetic field. In such a situation (called a cage radical pair), the electron spins precess and their singlet/triplet character can vary with time in a field-dependent manner. Thus, if a chemical reaction is more (or less) favorable in a singlet than in a triplet state, the overall reaction rate can depend on the ambient magnetic field, even for fields near earth-strength (about 0.5 G). Also, since there are many biological radical processes being found to play roles in development, aging, disease, and normal cell functioning, it seems natural to use the radical pair mechanism to approach biological magnetic field effects. While Chapters 1 and 2 elaborate on the above ideas, Chapters 3 to 5 examine the effects of oscillating magnetic fields on chemical reaction yields within the radical pair mechanism. They seek a general treatment that can apply to a combination of a steady field with multiple oscillating fields, all at different frequencies and orientations with respect to the steady field. Since this work is aimed toward biological effects of magnetic fields and, in particular, magnetic sensory mechanisms, it should be sufficient to develop a general treatment that can hold in the natural magnetic environment. This is because the natural magnetic field undergoes slight variations in magnitude and direction with time and location, and these variations can provide potentially useful information on direction, location, time of day, or season to animals capable of detecting them [13]. Thus, certain animals may have evolved sensitivity to the natural magnetic environment, and effects due to man-made magnetic environments, which have not existed long on an evolutionary time-scale, may only be incidental. For these reasons, much of this thesis focuses on effects that can be achieved with very weak steady fields or low-frequency, low-intensity oscillating fields, as occur in nature. Thus, since the geomagnetic field is typically composed of a steady field of 0.5 G and a spectrum of oscillating fields with field strengths well below 0.03 G (a fluctuation due to a very large magnetic storm) [13], in order to hold in the natural magnetic environment, the general treatment is restricted in that it must be able to handle very weak steady fields but enjoys the freedom that it can use perturbation theory to treat the effects of the oscillating fields. This use of perturbation theory should also be applicable to a number of man-made magnetic environments where the steady field is much larger than the oscillating fields. Thus, a general treatment based on perturbation theory should be useful in certain health effects studies as well. While Chapter 3 gives a perturbation expansion based on the Schrodinger equation, Chapter 4 discusses a similar expansion based on the Liouville equation. Both methods give the same results if the singlet and triplet states depopulate with equal rates (k_S=k_T) in the Liouville method and an exponential time dependence (with the time constant or cage lifetime tau) is assumed in the Schrodinger method. In this limit, where k_S=k_T=1/tau, the spin and cage dynamics decouple (here cage dynamics includes the effects of chemical reactions and diffusion). Nevertheless, the Liouville equation allows a more general treatment that includes the effects of coupled spin and cage dynamics. This allows one to treat more thoroughly systems expressed by kinetic equations. The Liouville formalism also allows a more natural description of more complicated effects such as rotation, spin relaxation, diffusion, and multi-step reaction kinetics (state diagrams). In addition to the increased rigor and generality of the Liouville equation approach, this method happens to yield a more reliable, compact, and efficient computer algorithm than does the Schrodinger approach. This new Liouville approach does not suffer from certain numerical problems arising in the Schrodinger approach such as occur when evaluating integrals (3.11) and (3.12) in Chapter 3 near omega=0, especially for higher orders of perturbation theory. It also does not suffer from the rapidly increasing computation times (see Tables 3.2, 3.3, and 4.1) needed for the Schrodinger approach, thus allowing much higher orders to be done in a reasonable time frame. The Schrodinger method does, however, have some advantages. First, it gives insightful analytical expressions (like eq.(3.24)) relating oscillating field effects to matrix elements, eigenfrequencies, applied frequencies, and lineshape functions. It also is easily adapted to allow time-dependent yield calculations (Appendix E). Lastly, it can treat a variety of cage dynamics cases such as the exponential or Noyes time-dependences. Next, Chapter 5 discusses several mixed perturbation methods that attempt to bridge the gap between the Liouville and Schrodinger formalisms, allowing k_S != k_T yields to be calculated as in the Liouville approach while preserving the Schrodinger approach's insightfulness and ability to calculate time-dependent yields. Then, following the perturbation method chapters, Chapter 6 discusses low-field and low-frequency effects in B12 systems. While these effects are interesting in their own right, they serve as excellent examples of effects due to state mixing and level repulsions, a quite general approach for discussing effects of time-independent fields. Finally, note that while much of this thesis has been previously published [4,5,6,7], an attempt has been made here to integrate the results into a coherent whole, and certain sections (for example, Chapter 5 and the Appendices, the latter of which are included to help tie together and further explain the results of Chapters 1-7) contain new results not previously published. Chapter 7 - Conclusion The preceding chapters have discussed many things: Chapter 1 introduced the area of biological magnetic field effects (such as animal homing and orientation and potential health effects of electromagnetic fields), discussed how free radicals play many important roles in biology (such as in normal cell functioning, development, aging, disease, and cancer), and pointed out how a free radical mechanism for explaining biological magnetic field effects seems reasonable; Chapter 2 introduced the radical pair mechanism and several standard methods for calculating its singlet-to-triplet yields (rotating frame, numerical integration, Schrodinger and Liouville formalisms, etc.); Chapters 3-5 described several new perturbation methods developed to calculate oscillating magnetic field effects on the yields (namely, the analytical and insightful Schrodinger perturbation method of Chapter 3, the efficient, reliable, and quite general Liouville perturbation method of Chapter 4, and the insightful mixed perturbation methods of Chapter 5 that bridge the gap between the Schrodinger and Liouville methods, all of which serve as cross-checks on each other for certain parameter ranges); and finally Chapter 6 used these methods to calculate interesting and perhaps biologically relevant earth-strength steady and oscillating magnetic field effects on the yields in coenzyme B12 radical pair systems. While the previous chapters have discussed the key results of this thesis, the following appendices attempt to fill in the gaps between and tie together these main results as well as point out some directions for future research. For example, Appendices A, B, C, and D give proofs of certain results cited in or related to the main thesis text; Appendix E tells how to obtain time-dependent yields by substituting time-dependent lineshape functions into the perturbation series of Chapters 3 and 5; Appendix F discusses how powder or phase averaging can simplify the perturbation series for certain oscillating field effects; Appendix G discusses via the Bloch equations and Redfield matrices how to include relaxation in the Liouville formalism; Appendix H elaborates on the level repulsion idea of Chapter 6 for more general eigenvectors and lineshape functions, discusses in more detail the heights and widths of repulsion-induced peaks, and points out some shortcuts for calculations; Appendix I combines the rotating frame with the level repulsion idea of Chapter 6 and Appendix H to explain in detail a number of oscillating field peaks shown in figures in Chapters 3 and 4; Appendix J uses the perturbation series of Chapter 3 to explain in detail a number of steady and oscillating field effects shown in plots in Chapters 3, 4, and 6; and Appendix K discusses several EPR simulation routines made to test the spin Hamiltonians used in the thesis calculations. It is hoped that the interested reader will find these appendices (and especially the example calculations of Appendices I and J) useful for understanding the rest of the thesis and stimulating in their own right. Finally, it is hoped that the reader will appreciate the important contributions made by this work toward treating oscillating magnetic field effects in radical pair systems, finding and explaining effects of earth-strength steady and oscillating fields on several different types of spin systems, and addressing the still controversial topic of biological magnetic field effects. It is also hoped that many of the interesting phenomena shown and described in this thesis will be explored experimentally in the near future. Copyright 1997,1998 Jeffrey Michael Canfield, All Rights Reserved for more information see http://ierc.med.uiuc.edu/canfield.html or http://www.biosci.ohio-state.edu/~jcanfld/