keywords: Cryptochrome , Radical pair, Spin-Orbit Coupling, Electric dipole moment, Open quantum system

Significance↩︎

The nature of the biological signal generated via the radical pair-based magnetoreception remains an open question in quantum biology. In this regard, we presented a model of magnetoreception in which we calculated the electric dipole moment within the radical pair (a measurable magnetic field-dependent quantity) even at physiological temperature conditions in the framework of quantum modeling. Our findings promote the understanding of the mechanism of radical pair and its associated signaling. In addition, these data pave the way for the advancement of biomimetic-based technologies.

1 Introduction↩︎

The physiological ability to perceive and detect invisible signals of the Earth’s magnetic field by an internal and innate compass is the basis of magnetoreception and magnetic navigation in various animals, for example the European robin as a migratory bird. The study to discover and understand this compass has been the subject of extensive interdisciplinary research for decades and continues [1]–[5]. Since the geomagnetic field is a very weak field that can penetrate biological systems [6] from a biophysical perspective, magnetic sense can be described as the interaction between the magnetic field and biological structure. To date, various experimental studies, including behavioral, histological, and electrophysiological, have been performed on diverse organisms, namely bacteria (magnetobacteria [7], [8]), invertebrates (such as beetles [9], [10] and honeybees [11]), and vertebrates (including birds [12]–[14], fish [15], and mice [16]–[18]) to investigate magnetoreceptors. In this regard, various theoretical and modeling studies have been conducted to better understand magnetic sense [19]–[25]. In general, based on these studies, two types of receptors have been proposed to be involved in the mechanism of magnetoreception. One type is an iron-based magnetoreceptor such as biogenic magnetite, which can be studied from a classical perspective. The other type is a magnetosensitive photoreceptor, whose radical pair mechanism is considered as a potential intersection between biology and quantum mechanics. In the mechanism, spin-correlated radical pairs are formed in flavoproteins such as cryptochromes (Cry) during a photon-absorption-dependent process. So far, six Cry types have been identified, with Cry4a and Cry1a being likely candidates for radical pair-based magnetoreception [26]. In this context, It is hypothesized that an external magnetic field can affect the spin dynamics of unpaired electrons via the Zeeman effect, which causes splitting and lifting of the degeneracy of triplet sublevels [12] and changes the interconversion rate between singlet and triplet spin states. Hence, most theoretical studies have only considered spin degrees of freedom in their models [20], [22], [27], [28]. It should be noted that in addition to its proposed role in magnetoreception, the radical pair mechanism also plays a role in other biological processes, namely reaction centers in photosynthesis [29], regulation of circadian rhythms in some organisms [30], and DNA repair [31]. For a radical pair to act as a magnetic compass, the interconversion between its spin states must be influenced by anisotropic interactions [32]. To our knowledge, two main sources of anisotropy are known. One arise from the magnetic coupling between the electron spin and the atomic nuclear spin [33], in which isotopic substitution [34] can change the strength and orientation of the hyperfine coupling and hence modulate the direction-dependent sensitivity, and the other originates from the spin-orbit coupling between the electron spin and its orbital angular momentum [35]. The radical pair mechanism aligns with the known features of the avian internal compass, including its function as a light-dependent inclination detector [24]. Various pathways viz-a-viz chemical and physical transduction, have been proposed to determine the mechanism by which an external magnetic field affects the radical pair, as follows. The change in the interconversion rate of singlet and triplet states induced by the magnetic field, and the subsequent change in the yield of spin states, results in a change in the yield of the chemical product [36]. It is predicted that the bird can detect this difference in field-dependent yield and use it for navigation. Another biochemical hypothesis supported by experimental evidence suggests that the Cry4a in birds interacts with retinal cone-specific G-proteins, potentially initiating a signaling cascade that alters cell membrane potential [26]. In contrast, a biophysical hypothesis proposes that a long-lived triplet radical pair forms in cryptochrome, possessing an electric dipole moment capable of generating an electric field. The field may be able to influence retinal protein isomerization [37], [38]. Based on what is briefly reported above, it can be concluded an unambiguous and definite final comment concerning radical pair-based mechanism and signal transduction has yet to be provided, but it seems that the biophysical transduction mechanism offers a simpler way for magnetic field detection.

Given magnetic sensitivity of radical pair system, the aim of this study is to provide, a method for either detecting magnetic fields or transducing them into measurable signals, within the framework of quantum mechanical calculations. For this purpose, the behavior of the system is investigated by considering both the dynamics and transitions related to the spin component and the spatial asymmetry and charge separation related to the spatial component via considering the spin-orbit coupling. Accordingly, the electric dipole moment is expected to emerge as a measurable physical quantity that has the potential to interact with biological structures. In this research, the effect of the characteristics of an external magnetic field on the behavior of the radical pair dipole moment is analyzed in detail. In fact, the considered approach, by integrating spin coherence and electrostatic asymmetry, provides a more complete view in this regard. It is worth noting that in addition to confirming the radical pair as a magnetic biosensor, the results of this study could play a role in the advancement of bio-inspired technologies which replicate animal magnetic sensitivity.

The organization of the paper is as follows. In Section 2, the relevant quantum model is provided. This is followed by the results of our numerical solution and a discussion in Section 3 and finally, the paper is concluded in Section 4.

2 Theoretical method↩︎

The absorption of electromagnetic waves, having specific intensity and wavelength, by the flavin adenine dinucleotide (FAD) chromophore can trigger a chain of electron transfer in cryptochrome protein that results in the formation of paired radical intermediates (Fig 1). Initially, the state of these spin-correlated radical pairs can be either antiparallel to one another (singlet) or parallel (triplet). The spins of the unpaired electrons can interconvert under the influence of factors such as weak magnetic fields (strong magnetic fields can disrupt the mechanism), and it is the general basis of cryptochrome-based magnetoreception. We begin our analysis by considering the hyperfine interaction, which can play an important role in the spin dynamics and magnetosensitivity of radical pairs. The interaction of the internal magnetic field caused by the magnetic moments of atomic nuclei with the unpaired electron is called the hyperfine interaction [36]. The directional asymmetry in the molecular environment is indicated by a diagonal anisotropic hyperfine tensor (\(\mathbf{A}= \mathrm{diag} (A_x, A_y, A_z)\)) [2]. In this numerical simulation, in order to simplify and reduce the computational complexity, we have regarded only a single nucleus with spin-½. Although considering multiple nuclei with spins can provide a more accurate and realistic representation, their effects on the overall behavior of the system are negligible. The Zeeman interaction describes the oscillations in the electron’s magnetic moment due to the interaction of an external magnetic field (such as the Earth’s magnetic field) with the spin of electrons [36]. The intensity of these oscillations depends on the strength of the magnetic field.

The Earth’s magnetic field in the absence of an oscillating field in spherical coordinates system is as follows: \[\mathbf{B} = B_0 (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)\] , which is represented in Figure 2. To simplify the analysis, assuming axial symmetry in the system, the angle \(\phi = 0\) is considered [2].

In general, the strength of the Zeeman interaction is related to the magnetic field strength and the gyromagnetic ratio (\(\gamma\)). Where \[\gamma = \frac{1}{2} \mu_B g_s\] determines the extent to which the electron’s magnetic moment couples to the external magnetic field. \(\mu_B\) is the Bohr magneton and \(g_s\) is the electron spin g-factor.

The time evolution of the considered system is described based on the von Neumann equation. \[\frac{d}{dt} \rho(t)= -\frac{i}{\hbar} [{H}, \rho] \label{von}\tag{1}\]

where \(\rho\) is the density matrix, and \(H\) is the Hamiltonian of the system. Considering the above and in the absence of inter-radical interactions [20], the spin Hamiltonian of the radical pair system, which includes hyperfine and Zeeman interactions, is expressed as follows:

\[{H} = \mathcal{I} \cdot A \cdot {S}_1 + \gamma B \cdot ({S}_1 + {S}_2) \label{Hamiltonian}\tag{2}\] where \(\mathcal{I}\) defines the nuclear spin operator, \(S_1\) and \(S_2\) are the spin operators of the two unpaired electrons in the radical pair. The ratio between the strength of the hyperfine and Zeeman interactions has significant effects on the quantum behavior of the radical pair system. Accordingly, if the ratio ( \(\frac{A_z}{\gamma B_0}\)) is small and in the range ( 0,\(2\times10^{-3}\)), the Zeeman interaction dominates, and if this ratio exceeds 3, the hyperfine regime will dominate. In conditions where this ratio is in the intermediate range between \(2\times10^{-3}\) and 3, neither interaction alone dominates and the effects of both factors are involved in the dynamics of the system [2]. The effects of the interaction between spin and spatial momentum are ignored in most common radical pair models. However, according to the Dirac equation, an electron moving in an electric field experiences an effective magnetic field [39], which leads to the formation of spin-orbit coupling (SOC). This interaction is of the fine-structure type and links the electron’s spatial position to its spin state. SOC is usually stronger than hyperfine interactions and makes the spin evolution dependent on molecular orbital symmetry. The interaction can play a notable role in spin dynamics, transitions between spin states, and magnetosensitivity even in the absence of nuclear spin [35], through entanglement between spatial and spin states. Accordingly, it seems necessary to consider spatial degrees of freedom in modeling radical pairs. Considering the spin-orbit coupling Hamiltonian as follows: \[H_{SOC}=\zeta_{SOC} L. S \label{SOC}\tag{3}\] the spin angular momentum of the electrons is coupled to their orbital angular momentum. Where \(L\) and \(S\) are the orbital angular momentum and spin operators respectively, and \(\zeta_{SOC}\) is the spin-orbit coupling constant. In this model, the orbital angular momentum is considered as a spin-1 system in a three-dimensional Hilbert space consisting of three sublevels with magnetic quantum numbers \(m_l = {-1, 0, +1}\). The strength of the coupling is determined by a coupling constant that depends on the electric field experienced by the electron and also on the electronic configuration of the molecule. Based on the magnitude of this constant, the spin-orbit coupling can be classified into one of three regimes: weak, moderate, or strong. Although the radical pairs in cryptochromes are made of organic molecules, the relatively large structure and the presence of \(\Pi\)-conjugated systems cause them to have significant SOC even without the presence of heavy elements. Now, considering the spin-orbit interaction, the Hamiltonian of the system can be written as below to obtain a more accurate description of the quantum dynamics of the radical pair.

\[{H} = \mathcal{I} \cdot A \cdot {S}_1 + \gamma B \cdot ({S}_1 + {S}_2) + \sum_{j=1}^{2} \zeta_j \, {L}_j \cdot {S}_j \label{FullHamiltonian}\tag{4}\] where \(j\) refers to each radical. In this model, the tensor product of a 2-level nuclear spin by two 2-level electron spins by two 3-level spatial momentum for each electron creates a 72-dimensional Hilbert space. The initial spin state of the radical pair is considered to be a singlet, defined as follows: \[\vert\psi_e\rangle=\frac{1}{\sqrt{2}} (\vert 0 1\rangle - \vert 1 0\rangle)\]

The initial state of the nuclear spin is considered to be a completely mixed state, which is described by the following density matrix: \[\rho_I= \frac{1}{2}(\left| \uparrow \right\rangle \left\langle \uparrow \right| + \left| \downarrow \right\rangle \left\langle \downarrow \right|)\]

The spatial component is considered as a two-level system that transitions from the ground state \(|g\rangle\) with \(L_z=-1\) to the excited state \(|e\rangle\) with \(L_z=0\). In this model, changes in orbital angular momentum can be probed in Hilbert space. The initial state of the spatial part of the system is defined as follows: \[\vert\psi_{spatial}\rangle= \frac{1}{2}(\vert g\rangle _1+ \vert e\rangle _1 )(\vert g\rangle _2+ \vert e\rangle _2 )\] we considered our initial state entangled in its spin states (singlet) and separated in spatial states ( Fig 1).. However, during the time evolution of the system, both the spatial and spin components become entangled. Our main goal in considering the interaction between electron spin dynamics and its spatial dynamics is to determine the biophysical mechanism for detecting or converting the Earth’s magnetic field information into understandable signals through the electric dipole moment.

We expect this dipole moment to be sensitive to the properties of the external magnetic field. On the other hand, the electric field resulting from the dipole moment may be able to interact with surrounding biological structures and thus establish a pathway for transmitting the necessary information for magnetic navigation. the dipole moment operator for the radical pair system is: \[\mathbf{p} = \sum_i \mathbf{e }\, \mathbf{r_i}\] Where \(\mathbf{e}\) is the elementary charge , and \(\mathbf{r}\) is the position operator. The vector operator \(\mathbf{r}\) is expressed as:

\[\mathbf{r} = \hat{x} \, {\imath} + \hat{y} \, {\jmath} + \hat{z} \, {k}\] where \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) are the position operators along the x, y, and z directions, respectively.

Generally, the position operator operates in a continuous and infinite-dimensional Hilbert space, but the radical pair system is modeled in bounded and spin-based Hilbert space, so continuous position operators cannot be used directly. Here, the components of the position operator were approximated using spin-state transitions and ladder operators. In other words, the x component of the position operator by the rotating wave approximation (RWA), the energy raising (\(\hat{\sigma}^+\)) and lowering (\(\hat{\sigma}^-\)) operators, is defined as follows:

\[\hat{x} = \sqrt{\frac{\hbar}{2 m \omega}} \left( \hat{\sigma}^+ + \hat{\sigma}^- \right)\] In addition, the position operators in the \(\hat{y}\) and \(\hat{z}\) directions can be expressed similarly, and finally the spatial behavior of spin transitions in a finite Hilbert space can be modeled.

Indeed, so far we have considered the radical pair as a closed and ideal system. But to describe the realistic behavior of a radical pair, we must regard environmental effects. Because radical pairs of cryptochrome protein are not isolated and interact with surrounding biomolecules and solvent at high physiological temperatures. These interactions cause decay of system coherence and destroy quantum properties of biological structure. Here, we have modeled the radical pair as an open quantum system that interacts with the environment through the Lindblad master equation [40] as follows: \[\frac{d}{dt} \rho_s = -\frac{i}{\hbar} [H_s, \rho_s] + \mathcal{L}(\rho_s)\] Where \({L}\) is the Lindblad superoperator, which is expressed as follows: \[\mathcal{L}(\rho_s) = \Gamma \sum_i \left( A^\dagger_i \rho_s A_i - \frac{1}{2} A^\dagger_i A_i\rho_s - \frac{1}{2} \rho_s A^\dagger_i A_i \right) \label{lin}\tag{5}\] Where \(\Gamma\) is dissipation rate, and \(A\) is the collapse operator ( \(A_1 = \sigma_1^- \otimes I_2\) and \(A_2 = I_1 \otimes \sigma_2^-\)). where \(I\) is identity matrix. Most studies on radical pair systems have only considered the interaction between the system and the environment via spin degrees of freedom. Noting that the environment can spatially impact biomolecules, it is more realistic and accurate to consider The coupling of the position of the system with that of the environmental particles. Accordingly, we have addressed this type of coupling in our research. Table 1 shows the values of model parameters.

3 Numerical Results and Discussion↩︎

linear multistep method (Adams method) is used for numerical solution of the von Neumann equation (Eq 1 ), to study the time evolution of the radical pair system. We considered using numerical approach due to the considerable complexity of the system and the size of the Hilbert space, which incorporates multiple quantum degrees of freedom. This structure enables the model to coherently capture spin dynamics, spatial asymmetry, and spin–orbit interactions as the key elements underlying the emergence and modulation of the electrical dipole moment under external magnetic field influence. The study of the evolution of the expectation value of the dipole moment (\(P_x\)) through time, involves two periods in the order of \(10^{-7}\) s and \(10^{-3}\) s ( Fig 3 ). This behaviour is expected as 72D Hamiltonian of the system has eigenvalues in the order of \(10^7\) and \(10^3\).

Since we understood the general behaviour of the system, we studied the evolution of the expectation value of the electrical dipole moment through time, in a variety of inclination angle of magnetic field lines (\(\theta\)). As is shown in Figure 4, the time evolution of dipole moment expectation value (\(P_x\)), shows a strong dependency to the value of the inclination angle. Moreover it seems the sensitivity to the changes of the inclination angle rises exponentially, very fast.

In general, as the bird moves, the orientation and magnitude of the magnetic field vector change with respect to the bird’s location. Accordingly, in this paper, we performed rather extensive computational modeling to elucidate the effects of the magnetic field’s inclination angle, \(\theta\), and magnitude on the quantum dynamics of the radical pairs. Firstly, we investigated the changes in the electric dipole moment with respect to the magnetic field’s inclination angle. The analysis was carried out in the order of microseconds, and the system was considered to be closed and coherent during this time. Due to the lack of environmental interactions and dissipative effects in the ideal framework considered, coherence is maintained all through time evolution. It should be noted that the effects of decoherence under physiological temperature have been investigated in Subsection 3.1. Figure 5a shows the variation of the x-component of the electric dipole moment, \(P_x\), for different angles of the magnetic field. The results reveal a notable and periodic dependence on \(\theta\). Furthermore, identical values at \(\theta = 0\), \(\pi\), and \(2\pi\) reflect that although the system is sensitive to the inclination angle, it does not detect polarity of the magnetic field.This finding is consistent with reported observations [41]. Repeating this analysis at later times—such as 10 microseconds in Figure 5b—presents that the angular dependence of \(P_x\) remains qualitatively stable. generally, The radical pairs form and evolve in a narrow time window in the picosecond range [42]. During this time, the electric dipole associated with them can be accessible to the downstream biological structures. Also, quantum coherence is maintained at this ultrafast time scale. On the other hand, the response time and signal processing time in these structures may be much longer than the quantum coherence time (e.g., on the order of milliseconds), but given the average flight speed of the bird, it is expected that there will be no noticeable change in its position during this time frame. Consequently, it can be assumed that the magnetic field experienced by the radical pair system remains constant during this time, and the directional information carried by the electric dipole moment can be used to navigate the bird. It is expected that the radical pair in cryptochrome can only be considered a magnetic receptor if it is capable of participating in signal transduction processes [26]. The electrical signal resulting from the electric dipole moment related to radical pair may be transmitted through changes in the current of ion channels and cell transmembrane potential or changes in the conformation of cryptochrome-interacting proteins such as G-proteins [26] and ultimately interpreted in a specific part of the brain. Although it is an initial hypothesis, it is evident that further research, both experimental and computational/theoretical, is required in this important field, and our findings in this study could pave the way.

Then, following prior evidence [43] that radical pair dynamics in cryptochrome can be influenced by magnetic field strength, we evaluated the changes in the electric dipole moment with respect to the intensity of the external magnetic field for three different angles at one microsecond. As can be seen in Figure 6, at lower intensities, especially in the geomagnetic range (approximately \(25\)–\(65\mu T\)), there is an oscillatory and nonlinear dependence on \(B_0\), such that small variations in the field intensity lead to significant changes in the dipole moment amplitude. At intensities beyond the geomagnetic window, a gradual decrease in the dipole moment amplitude is revealed, and as a result, the system is less responsive to changes in the field strength. This finding is in agreement with a report that indicates that exposure to fields stronger than the geomagnetic field disrupts the orientation ability of birds.

We have generated a three-dimensional plot integrating the magnetic field properties, i.e., different angles and intensities, to show how the x-component of the electric dipole moment, \(P_x\), responds simultaneously to changes in the magnetic field inclination angle \(\theta\) and the strength \(B_0\). Since these properties can vary with latitude, it can be seen from Figure 7 that the \(P_x\) has a periodic dependence on the angle \(\theta\) as well as an obvious sensitivity to the magnetic field strength \(B_0\). This dual dependence of the dipole moment produces a response that causes the considered mechanism to act as more than just a compass. The mechanism effectively constitutes a quantum-based magnetic map that can indicate both direction and location. Consequently, radical pair-based magnetoreception may act not only as a navigator, indicating direction of movement, but also as a magnetic GPS, determining position in the geomagnetic field. Our findings support the hypothesis that the avian sensory system can interpret changes in the electric dipole moment to directional and spatial information for magnetic navigation.

3.1 Dissipative effect↩︎

A Variable-Step,Variable-Order method was used for the numerical solution of the Lindblad master equation Eq.@eq:lin to study the time evolution of the radical pair system under dissipative effect. Noting that the physiological temperature plays a key role in the dynamics of open quantum systems by changing the rate of dissipative processes, this effect is implicit in the \(\Gamma\) parameter of the master equation [44]. In the following, the dissipation role in the time evolution of the x-component of the electric dipole moment for various magnetic field inclination angles was investigated. As can be seen in Figure 8, the system’s behavior in the presence of environmental dissipation demonstrates angular dependence on the external magnetic field. Based on this Figure, in early times, \(P_x\) increase that indicate a strong coherent behavior and then exponentially decay which related to decoherence phase induced through environmental interactions at \(\theta=\pi\). When the inclination angle decreases—for instance, at \(\theta=\pi/2\) damped oscillations are exhibited, reflecting the dissipative effect, and at smaller angles, such as \(\theta=\pi/3\) and \(\theta=\pi/2\), irregular oscillations are seen, describing the system transitions into a less coherent state.

It is anticipated that the dissipative effect induced by thermal noise causes a decrease in the quantum behavior of biological systems over time, but according to Figure 9, it can be seen that when the radical pair system is subjected to dissipative effects and the system meets a stable state (steady-state regime), the changes in the electric dipole moment (\(P_x\)) are still angle-dependent. The data clearly shows that, despite the decoherence effect, the quantum mechanism still works in the radical pair system. It is interesting to note that our achievement in the magnetoreception field can challenge the common criticism of quantum biology, namely the loss of quantum properties of biological structures arising from high physiological temperatures.

4 Concluding remarks↩︎

In this study, we attempted to gain a more detailed understanding of the behavior of radical pairs under the influence of a magnetic field using quantum modeling. The effect of the spatial component on the system dynamics has been incorporated in the model by considering the spin-orbit coupling term in the Hamiltonian. The variations in the physical quantity of the radical pair induced by the application of static magnetic field has been investigated computationally. We have observed that both the inclination angle and the magnitude of the applied magnetic field have an obvious effect on the electric dipole moment of the radical pair. Furthermore, when the angle and intensity are changed simultaneously, the results showed that the system can provide both direction and position information. It is worth mentioning that the magnetosensitivity of the radical pair was maintained even when dissipation and decoherence effects were taken into account, which is a valuable achievement not only in the field of magnetoreception but also in quantum biology. These findings support the role of the radical pair system as a magnetic receptor and sensor in physiological conditions. These deta can have significant consequences in such areas as GPS-free navigation, magnetic detection of biomarkers, and assessment of electromagnetic pollution.

Authors’ contributions↩︎

A.R.B and M.S proposed the research problem and the model system; A.S mathematically formulated the problem and computed the results; S.A, N. H, S. R and A.B analyzed the data; M.S wrote the first draft, and all authors edited the final manuscript.

Declaration of AI use↩︎

AI (ChatGPT) was used solely for language polishing, and partially used in generation of schematic figures 1 and 2. The final manuscript was reviewed, revised, and approved by all authors, who take full responsibility for its content.

Competing interests↩︎

The authors declare no conflict of interest.

References↩︎