Simulation of C. elegans thermotactic behavior in a linear thermal gradient using a simple phenomenological motility model
Introduction
The nematode Caenorhabditis elegans has been reported to show thermotaxis, a sophisticated behavioral response to temperature. Hedgecock and Russell (1975) first found that C. elegans migrates to its cultivation temperature. Later, based on observations of genetically mutated worms and worms in which several neurons had been destroyed by laser ablation, Mori and Ohshima (1995) first suggested a neuronal mechanism for the regulation of thermotactic behavior. Since then, extensive research has been done to identify the genes required for regulation of thermotactic behavior in the nervous system of C. elegans (e.g., Mori and Ohshima, 1995, Mori, 1999, Ito et al., 2006).
However, there appears to be inconsistency among previous findings related to C. elegans thermotaxis. Early studies, such as those of Hedgecock and Russell (1975) and Mori and Ohshima (1995), suggested that wild-type (normal) C. elegans tends to navigate up or down thermal gradients to the region of its cultivation temperature from nearby regions. This tendency has been attributed to a counter-balancing of cryophilic drive and thermophilic drive in the C. elegans nervous system. On the other hand, Ryu and Samuel (2002) observed that individual C. elegans worms modulate their rates of abrupt reorientation when the temperature is higher than the cultivation temperature, so that they tend to reach regions of lower temperature. However, no evidence has been found to support the theory that C. elegans actively modulates its locomotion when the temperature is lower than its prior cultivation temperature. Yamada and Ohshima (2003) also reported that they observed warmth avoidance in C. elegans but found no evidence of thermophilic drive. Isothermal tracking, a tendency to track the isotherm when C. elegans is close to regions of prior cultivation temperature, has been known to exist for many years (Hedgecock and Russell, 1975). Luo et al. (2006) recently revealed several characteristics of isothermal tracking, including the distribution of its total duration time and mean duration time. They characterized isothermal tracking as a period during which large bending turns are suppressed. Observing the intracellular calcium concentration of AFD sensory neurons during temperature change, Clark et al. (2006) found the calcium concentration to be related to ambient temperature change above the prior cultivation temperature of the worms. They suggested that information from AFD neurons would be adequate for the known thermal response of individual worms with respect to cryophilic behavior above cultivation temperature and isothermal tracking.
Ito et al. (2006) observed that C. elegans populations positioned in a region cooler than their previous cultivation tended to navigate up thermal gradients to their cultivation temperature. However, this tendency did not become clear until 30 min or more after initiation of the test. Ito et al. argued that this delay in response would have made the detection of any thermophilic tendencies difficult. They proposed that the delay might be attributable to differences in transmission efficiency in the neural pathways through which cryophilic or thermophilic drives are conducted and, in particular, to the different numbers of synaptic connections in these pathways. White et al. (1986) studied the synaptic connections in the neuronal network of C. elegans extensively. However, details about the functions and the mechanism of synaptic interaction among neurons in these pathways remain largely unknown.
In the present study, we focused on the previously recognized cryophilic tendencies, because, although the existence of a thermophilic bias has not yet been ruled out, the thermophilic bias might not be a robust behavior. We hypothesized that if we could evaluate the effects of known individual thermotactic behavior, isothermal tracking and cryophilic tendencies induced by turn frequency modulation, we might obtain insight into the apparent inconsistency among previous observations. One motivation for the present study was the random walk property of C. elegans locomotion. The movement of C. elegans is not directed toward the region of the cultivation temperature but rather employs a somewhat passive biased random walk strategy (Ryu and Samuel, 2002, Clark et al., 2007). To test our hypothesis, we constructed a phenomenological mobility model to imitate individual thermotactic behavior. In the case of no thermal gradients and no boundary condition, we were able to obtain analytical results. However, in presence of thermal gradients, we found it very difficult to obtain analytical results and so carried out a Monte-Carlo style simulation using this model. As a result of the random walk property in the locomotion of individual worms, we found that the cryophilic tendencies can explain population-level thermotactic behavior to some extent and isothermal tracking can strengthen the population-level thermotactic behavior.
Section snippets
Methods
We constructed a simple phenomenological motility model of individual worms based on previous experimental results. For the case in which no thermal gradients were present and no boundaries were assumed, we could obtain analytical results. For the cases in which thermal gradients were present, we carried out a Monte-Carlo style population-level simulation, in which model worms migrate freely on a semi-infinite two-dimensional plane that was bounded only in the X-direction. The populations of
Analysis of the no-thermal-gradient case
Before beginning numerical simulation with the model described in Section 2.2.5, we attempt to obtain analytical results. If the thermal gradient is not present, then the locomotion of the model worm is a symmetric random walk. In this case, for no boundary condition, we can obtain analytical results.
It is expected that the distribution of the worms will become similar to a Gaussian distribution as time passes. One way to confirm this would be that, from Eq. (2), the trajectories of the model
Discussion
In the present paper, we investigated the thermotactic behavior of C. elegans, focusing primarily on the relationship between the individual and population thermal responses of worms. Based on previously published data on the individual thermal response of C. elegans (Ryu and Samuel, 2002, Luo et al., 2006, Clark et al., 2007), we constructed a simple stochastic individual model and performed a Monte-Carlo style numerical simulation using this model. The proposed model was too simple for the
Acknowledgments
The present study was supported by a Grant-in-Aid through the 21st Century Center of Excellence Program by the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by a Grant-in-Aid for Academic Group Research Project from Iwate University. The authors would like to thank Dr. Tokumitsu Wakabayashi for his useful discussion on improving our manuscript. The authors would also like to thank Dr. Masahiro Kanai of University of Tokyo for his personal communication on ZRP.
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