The establishment and interpretation of the concentration distribution of the morphogen Bicoid (BCD) is considered crucial for the successful embryonic development of fruit flies. However, the biophysical mechanisms behind the timely formation and subsequent interpretation of the BCD morphogen by its target genes are not yet completely understood. Here a discrete-time, one-dimensional quantum walk model of BCD gradient formation is used to explain both the observed values of diffusivity and its precise interpretation. It is shown that the decoding of positional information from the BCD morphogen by its primary target gene hb, with the observed precision of ∼ 10%, takes a time period of less than a second, as expected on the basis of recent experimental observations. From this the on-rate ( k on ) for the binding of BCD to its target loci is obtained. Furthermore, the model is also used to explain certain key observations of recent optogenetic experiments concerning the time windows for BCD interpretation. Finally, it is argued that the presented model represents a significant step in the utilization of quantum computation-based techniques in studying the dynamics of biological systems in general and in the field of developmental biophysics in particular.
The establishment and interpretation of the concentration gradient of the morphogen Bicoid ( Bcd ) is crucial for the successful embryonic development of fruit flies. However, the biophysical mechanisms behind the timely formation and subsequent interpretation of this prototypical morphogen gradient by its target genes are not yet completely understood. Recently a discrete time, one-dimensional quantum walk model of Bcd gradient formation has been successfully used to explain the observed multiple dynamic modes of the system. However, the question of its precise interpretation by its primary target gene hunchback ( hb ) remains still unanswered. In this paper it will be shown that the interpretation of the Bcd gradient by its primary target gene hb , with the observed precision of ∼ 10%, takes a time period of less than a second, as expected on the basis of recent experimental observations. Furthermore, the quantum walk model is also used to explain certain key observations of recent optogenetic experiments concerning the time windows for Bcd interpretation. Finally, it is concluded that the incorporation of quantum effects into the treatment of Bcd gradient represents a viable step in exploring its dynamics.
The problem of cell fate determination by morphogen gradients in the embryonic development of many multicellular organisms has been a long-standing and important one in developmental biophysics and dynamics. The first mathematical model was proposed by Francis Crick over 50 years ago by postulating a reaction-diffusion based mechanism underlying the whole process. The first real world morphogen, named Bicoid (Bcd), was identified by molecular biologists in late 1980s in the embryo of fruit fly Drosophila melanogaster . Subsequently, Crick’s classical random walk based model was used by biophysicists to explain the formation of Bicoid and other morphogen gradients. Very recently Fluorescence correlation spectroscopy (FCS) studies have revealed multiple modes of Bcd transport at different spatial and temporal locations across the embryo of Drosophila melanogaster. It has been be shown that these observations are best fitted by a model based on quantum mechanics. In such a model it is hypothesized that the transitory quantum coherences in collaboration with unitary noise are responsible for the observed dynamics and relaxation to a non-equilibrium steady-state of the Bcd morphogen gradient. In this paper, in addition to further clarifying the mathematical details underlying the quantum-classical model, we use the said model to explain the observed Bcd interpretation time by its primary target gene Hunchback ( hb ).
Fast computational algorithms are in constant demand, and their development has been driven by advances such as quantum speedup and classical acceleration. This paper intends to study search algorithms based on quantum walks in quantum computation and sampling algorithms based on Langevin dynamics in classical computation. On the quantum side, quantum walk-based search algorithms can achieve quadratic speedups over their classical counterparts. In classical computation, a substantial body of work has focused on gradient acceleration, with gradient-adjusted algorithms derived from underdamped Langevin dynamics providing quadratic acceleration over conventional Langevin algorithms.
Since both search and sampling algorithms are designed to address learning tasks, we study learning relationship between coined quantum walks and underdamped Langevin dynamics. Specifically, we show that, in terms of the Le Cam deficiency distance, a quantum walk with randomization is asymptotically equivalent to underdamped Langevin dynamics, whereas the quantum walk without randomization is not asymptotically equivalent due to its high-frequency oscillatory behavior. We further discuss the implications of these equivalence and nonequivalence results for the computational and inferential properties of the associated algorithms in machine learning tasks. Our findings offer new insight into the relationship between quantum walks and underdamped Langevin dynamics, as well as the intrinsic mechanisms underlying quantum speedup and classical gradient acceleration.