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Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 76, Issue 6, 3026-3030, 1 June 1999

doi:10.1016/S0006-3495(99)77455-2

Biophysical Theory and Modeling

Biological Effects Due to Weak Electric and Magnetic Fields: The Temperature Variation Threshold

James C. Weaver^,  , Timothy E. Vaughan and Gregory T. Martin

Harvard–M.I.T. Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA

Address reprint requests to Dr. James C. Weaver, Harvard–M.I.T. Division of Health Sciences and Technology, Room 16-319, Massachusetts Institute of Technology, Cambridge, MA 02139. Tel.: 617-253-4194; Fax: 617-253-2514.

Weak fields are incapable of directly breaking chemical bonds (Valberg et al). Thus, if weak fields are the basis of biological effects, these can only occur by the alteration of ongoing biochemical reaction or transport processes (Astumian et al,Weaver et al). But the rates of these processes depend significantly on temperature. To illustrate consequences of realistic temperature variations, we consider a single process with rate

(1)

A more general formulation uses stochastic resonance with a noise density D≥kT (Astumian et al). The temperature sensitivity is often described by a first-order coefficient, α_T=(1/J)(∂J/∂T), with α_T=U₀/kT² for Eq. (1), but using only the Bolzmann factor. Typically U₀≈8kT, so α_T≈0.03°C⁻¹. The net temperature coefficient of more complex, multiple rate processes can be measured; in some cases it is very small or even negative, because of entropic effects, but it is seldom zero.

Biological systems experience significant temperature variations. Human core body temperature undergoes daily variations greater than 1°C (Hammel, 1968,Rubin, 1987,Keatinge et al,Shiraki et al,Webb, 1992), with larger variations in the extremities. In vitro electric and magnetic field experiments use feedback control (e.g., temperature-regulated exposure chambers), with variations greater than ∼0.01°C for exposure times of more than ∼100s (Appendix A ). Significantly smaller in vitro variations (e.g., ±0.002°C) are achieved only with nontrivial effort (Fulton et al).

Temperature variations cause molecular changes by varying J, which together with fundamental stochastic fluctuations compete with the molecular change due to the field exposure. This competition defines a lower bound to a response threshold. For zero field and steady temperature, an ongoing biochemical process is presumed to proceed at a quasisteady (basal) rate J₀, with a molecular change after an exposure time t_exp of ₀ ≈ J₀t_exp. This is the average number of molecules passing through a biochemical pathway or accumulating at an end point.

A weak field alters J slightly, via a biophysical mechanism such as voltage-gated cell membrane proteins (Astumian et al,Weaver et al), creating an additional field-induced molecular change:

(2)

(3)

(4)

Changes in both systems have stochastic variations approximately equal to , but they differ by _S, the molecular change due to the field exposure. Thus

(5)

(6)

(7)

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Figure 1

The most important parameters in the temperature history of the exposed and control biological systems are the time-averaged temperature, T_dc,exp and T_dc,con, followed by the magnitude of a sinusoidal temperature variation representing temperature regulation, T_ac,exp and T_ac,con, respectively. The complete parameterization is

where the terms in brackets indicate quantities with random errors of the magnitude indicated. Our estimates are conservative in the sense that the systematic temperature difference between the systems is zero (T_dc,exp=T_dc,con). We also neglect differences in the sinusoid periods, phases, and other parameters (e.g., slight differences in exposure time), whose random distribution would further increase temperature variability (details in Appendix B ).

Display large version of this figure

Figure 2

Dependence of R_thresh, the ratio of the temperature variation threshold to the fundamental molecular shot noise threshold, on barrier height, for a biochemical process described by Eq. (1). The dark-shaded band corresponds to in vivo temperature variations, with ΔT_dc=1–3°C; the light-shaded band corresponds to in vitro temperature variations, with ΔT_dc=0.01–0.1°C. This result is independent of the biophysical mechanism (voltage-gated channels, radical pair reactions, magnetite-medicated processes, etc.) involved in coupling a weak 50–60-Hz field to an ongoing biochemical process. Typical processes haveU₀≈8kT. Those with large U₀ have high fundamental thresholds, whereas those with small U₀ depart from “biological conditions” in that the spontaneous rates are large and, in the absence of a rate-limiting barrier, lack biological control (Weaver et al).

Fig. 3 further illustrates the importance of temperature variations by estimating the electric and magnetic field thresholds for the biophysical mechanism of voltage-gated membrane channels in a long cell (L_cell=1mm) (Weaver et al). The lowest curve (dot-dashed line independent of U₀) arises from thermal (Johnson-Nyquist) noise due to physical considerations only (Weaver and Astumian, 1990), with a threshold E_min≈2×10⁻⁶V/cm for a small bandwidth (Δf=100Hz) (Robertson and Astumian, 1991).

Display large version of this figure

Figure 3

Predicted threshold electric and magnetic fields for voltage-gated channels in a long cell for a prolonged exposure (t_exp=10⁴s) relevant to weak 50–60-Hz fields (Weaver et al). This estimate is based on the predicted molecular (ionic) change due to a two-state channel model that competes with molecular changes (N and V); in real cells other processes (e.g., initiation of action potentials, electroporation) would occur at the larger electric field values and prevent the molecular change of a long exposure from occurring. The magnetic field case assumes an induced electric field using a current loop radius relevant to humans, r_loop=0.3m. The upper two bands indicate the temperature variation threshold, with the dark-shaded in vivo band assuming ΔT_dc=1–3°C and the light-shaded in vitro band having ΔT_dc=0.01–0.1°C. The “molecular shot noise” curve (dashed line) is based solely on fundamental molecular change fluctuations (Weaver et al). The lowest curve (dot-dashed line) does not involve molecular changes, but instead arises from Johnson-Nyquist noise, which involves purely physical quantities (Weaver and Astumian, 1990).

Equation (4) yields the next higher threshold curve (dashed line) (Astumian et al,Weaver et al), which for the long cell yields E_min as a function of U₀ (the smallest value is E_min≈10⁻⁴V/cm; Weaver et al) and is about two orders of magnitude higher. The temperature variation threshold is represented by the top two bands, which are much higher than the fundamental molecular shot noise limit. These two bands are based on Eq. (7), including dc and ac temperature variations. The lower (light-shaded) band gives in vitro thresholds; the upper (dark-shaded) is the range for humans in vivo.

Table 1 shows a comparison between the predicted temperature variation threshold and measured electric field thresholds from in vitro experiments (Serpersu and Tsong, 1983,Graziana et al) on relatively simple biological systems, “voltage-gated” membrane-(Na,K)ATPase, where U₀≈5kT. In both experiments, the observed thresholds are consistent with the predicted temperature variation threshold for ΔT_dc in the range of 0.01–0.1°C.

Biological sensory systems could escape the temperature variation threshold in two ways. First, evolutionary pressure may result in biochemical temperature compensation, for example, by utilizing two biochemical rates in series, each with nearly the same temperature coefficient. This is analogous to providing electrical circuit temperature compensation by using a voltage divider with matched elements. Second, neural processing may be involved to correct for sensed temperature variations.

For a nonsensory system (such as unorganized cells in vitro) to respond, however, field-induced changes must exceed the temperature variation threshold. Thus, if observed nonsensory effects are indeed due to weak 50–60-Hz field exposures in the presence of typical temperature sensitivities and typical temperature variations, then _S ≥ Δn_V, which implies involvement of an extraordinary biophysical mechanism, one that couples to the biochemical process orders of magnitude more strongly than voltage-gated membrane channels in a large cell (Fig. 3).

Moreover, in vitro conditions are artificially “quiet,” in the sense that both temperature variations and other sources of molecular change competition are smaller than in in vivo conditions, or are absent altogether. This may allow in vitro observation of changes due to weak, extremely low frequency fields. For example, an impressive experiment using fibroblasts within a collagen matrix reported biochemical synthesis changes due to weak electric fields (McLeod et al). A strong coupling between the field and the cells may be provided through electromechanical deformation of the charged extracellular matrix and cells, observable in vitro because normal tissue movement and associated molecular changes are absent (Vaughan and Weaver, 1998), as are normal physiological variations in regulatory biochemical levels. Not only are both N and V important, but also V_{in vivo}>V_{in vitro}, and other competing molecular changes further constrain the ability of weak 50–60-Hz fields to cause biological effects in vivo.

If an observed effect is to be convincingly interpreted as being due to an electric or magnetic field, then the apparently overwhelming molecular change due to realistic temperature variations must be understood. This requires controls that determine 1) the temperature sensitivity of the relevant biochemical process (e.g., biochemical synthesis, cell growth, enzyme activity, receptor binding) and 2) the order-of-magnitude temperature variations within the biological system. In most reports of weak 50–60-Hz effects, however, these controls are absent.

Reconciliation of reported in vitro observations is particularly challenging, as these experiments involve relatively unorganized cellular systems in comparison to the evolved multicellular systems that are believed to underlie electric and magnetic sensory systems. Sensory systems have a further advantage: they often respond in short times in which temperature variations are minimal and stochastic resonance can be of greatest benefit. However, for nonsensory systems and long exposures (both in vivo and in vitro), temperature variations are larger and stochastic resonance is ineffective. The suggestion (Bezrukov and Vodyanoy, 1997a,Bezrukov and Vodyanoy, 1997b) that single cells might use stochastic resonance over very long times (up to 10⁶ s) therefore appears unrealistic. Without an explicit analysis of competing molecular changes due to temperature variations, it is difficult to accept reports of effects associated with weak 50–60-Hz fields as being caused by these very small physical stimuli.

Appendix A

Heat transfer by conduction is diffusive, damping regulating temperature variations. We represent temperature regulation as a localized sinusoidal temperature source, with frequency f_T≈10⁻²Hz. At a distance x away from the source, sinusoidal temperature variations of frequency f_T are reduced by a factor R_damp ≈ exp(), where κ is the thermal diffusivity. Typically, κ≈10⁻³cm²/s.

Active cells in vivo are typically within x≈20μm from a blood vessel, yielding R_damp≈0.99 (damping of 1%). Therefore, these cells experience essentially the full temperature variation of circulating blood, which in the core of humans varies by ≥1°C, and elsewhere is larger. This motivates our choice of the range 1–3°C for the in vivo temperature variation magnitude for exposure times greater than 100s.

In contrast, in vitro values of x typically range from 0.1 to 1cm, yielding 0.04–0.7 for R_damp. These values are consistent with microdegree ac variations in the bulk of a sample for short-duration (seconds) in vitro experiments. Long-duration (many minutes) experiments, however, will be subjected to significant dc temperature differences, but these are usually smaller than in in vivo systems; we use variations in the range 0.01≤(ΔT)_dc≤1°C to represent average, random dc temperature variations in vitro.

Appendix B

Competing changes arise from slight average (offset) random temperature differences ΔT_dc that exist between the control and exposed systems (Fig. 1). Typically, ΔT_dc≥0.01°C in vitro, and ΔT_dc≥1°C in vivo. The associated average molecular change difference is

(B.1)

A sinusoidal (ac) temperature variation provides a simplified representation of temperature regulation. The sinusoidal contribution is obtained by using a second-order expansion of J(T) around T₀≈310K:

(B.2)

For U₀=8kT and T=310K, β_T≈2.5×10⁻⁴K⁻². The nonlinear temperature dependence of the Bolzmann factor in Eq. (1) results in rectification for an ac temperature variation at f_T≪1Hz. The cumulative average molecular change is

(B.3)

If regulation were perfect, then each system would experience this additional molecular change _V,ac, and the difference between the two systems would remain unchanged. However, if there are slight, random differences in temperature regulation, the resulting molecular change variability further obscures _S. For brevity, we consider only one source of ac temperature variability: slight differences in ΔT_ac, the peak ac temperature, T_ac (Fig. 1). This yields fluctuations in molecular change,

(B.4)

For independent dc and ac temperature variations, the associated total molecular change is

(B.5)

which is used in Eq. (7) to determine the temperature variation threshold (involving both shot noise and temperature variability, with the latter dominant; see Fig. 3).