A Gravitational Theory of the Quantum

The synthesis of quantum and gravitational physics is sought through a finite, realistic, locally causal theory where gravity plays a vital role not only during decoherent measurement but also during non-decoherent unitary evolution. Invariant set theory is built on geometric properties of a compact fractal-like subset $I_U$ of cosmological state space on which the universe is assumed to evolve and from which the laws of physics are assumed to derive. Consistent with the primacy of $I_U$, a non-Euclidean (and hence non-classical) state-space metric $g_p$ is defined, related to the $p$-adic metric of number theory where $p$ is a large but finite Pythagorean prime. Uncertain states on $I_U$ are described using complex Hilbert states, but only if their squared amplitudes are rational and corresponding complex phase angles are rational multiples of $2 \pi$. Such Hilbert states are necessarily $g_p$-distant from states with either irrational squared amplitudes or irrational phase angles. The gappy fractal nature of $I_U$ accounts for quantum complementarity and is characterised numerically by a generic number-theoretic incommensurateness between rational angles and rational cosines of angles. The Bell inequality, whose violation would be inconsistent with local realism, is shown to be $g_p$-distant from all forms of the inequality that are violated in any finite-precision experiment. The delayed-choice paradox is resolved through the computational irreducibility of $I_U$. The Schr\"odinger and Dirac equations describe evolution on $I_U$ in the singular limit at $p=\infty$. By contrast, an extension of the Einstein field equations on $I_U$ is proposed which reduces smoothly to general relativity as $p \rightarrow \infty$. Novel proposals for the dark universe and the elimination of classical space-time singularities are given and experimental implications outlined

Summary of changes in repetitive element expression detected by microarray repeat-annotation in this study.

<p>Statistically significant upregulation and downregulation of repetitive element expression in mutant ES cell lines or testes is indicated by up and down arrows respectively. Changes that only appear to affect a small number of probes for a repetitive element are indicated in brackets. The degree of change in gene expression detected for these elements is detailed in the main text.</p>*<p>Although changes in <i>MMERVK10C</i> expression were not detected in Dnmt TKO ES cell microrray data in this study, RNA-seq analysis suggests that some genomic copies of <i>MMERVK10C</i> are upregulated in Dnmt TKO ES cells <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486-Karimi1" target="_blank">[22]</a>.</p

LTR retrotransposon targets of polycomb repressive complexes in ES cells.

<p>(A, C, E) MA-plots for <i>Ring1B^−/− Eed^−/−</i> double knockout, <i>Ring1B^−/−</i> single knockout and <i>Eed^−/−</i> single knockout <i>ES</i> cells showing how different classes of LTR retrotransposons change expression in these cell lines. (B, D, F) Plots showing the behaviour of selected retrotransposon probe populations in <i>Ring1B^−/− Eed^−/−</i> double knockout, <i>Ring1B^−/−</i> single knockout and <i>Eed^−/−</i> single knockout <i>ES</i> cells. The selected retrotransposons are all represented by multiple upregulated probes (≥4 fold upregulation, p<0.01) in <i>Ring1B^−/− Eed^−/−</i> ES cells. Vertical lines indicate a 4 fold change. Note that some retrotransposons (e.g. <i>MMVL30</i>, <i>RLTR45</i>) are upregulated in double knockout but not single knockout ES cells, other retrotransposons (e.g. <i>RLTR44</i>) are upregulated in all three ES cell lines. Retrotransposon probes are colour-coded as shown in the plot legends.</p

Closely related retrotransposons are differentially sensitive to loss of <i>Tex19.1</i>.

<p>(A) Phylogeny of mouse retrotransposon pol and pro proteins. <i>MMERVK10C</i> sequences are highlighted in red. The <i>MMERVK10C</i> sequences lie within a cluster of <i>IAP</i>-type sequences (yellow). (B) Plot showing the likelihood of <i>IAP</i> probes changing expression in the <i>Tex19.1^−/−</i> microarray dataset. (C) qRT-PCR for retrotransposons closely related to <i>MMERVK10C</i> in <i>Tex19.1^−/−</i> knockout and littermate control testes at 16 dpp. Expression levels for each repetitive element (mean ± standard error for three animals) were normalized to <i>β-Actin</i> and expressed relative to littermate controls. MMERVK10C env.c and IAP primer sets (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486.s007" target="_blank">Figure S2</a>) were used to assess <i>MMERVK10C</i> and <i>IAPEz</i> expression. Asterisk indicates a statistically significant difference (p<<i>0.05</i>).</p

Hdac1 regulates expression of LTR retrotransposons in mouse ES cells.

<p>(A) qRT-PCR verification of <i>LINE-1</i>, <i>RLTR45</i> and <i>IAP</i> expression in <i>Hdac1^−/−</i> ES cells. Expression levels (mean ± standard error for three biological replicates) were normalized to <i>β-Actin</i> and expressed relative to control ES cells. IAP and LINE1 5′UTR primer sets (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486.s007" target="_blank">Figure S2</a>) were used to assess <i>IAP</i> and <i>LINE-1</i> expression. Asterisks indicate a statistically significant difference (p<<i>0.05</i>) for <i>RLTR45</i> and <i>IAP</i> elements. <i>RLTR45</i> expression is upregulated in <i>Hdac1^−/−</i> ES cells, but IAP expression is downregulated. (B) Enrichment of LTR retrotransposon sequences in Hdac1 ChIP-seq data from mouse ES cells. The maximum likelihood of enrichment (±95% confidence intervals) for <i>RLTR45</i> LTR and <i>IAP</i> LTR sequences Hdac1 ChIP-seq relative to whole cell extract is shown. <i>RLTR45</i> LTR sequences are enriched in the Hdac1 ChIP-seq indicating a physical association between Hdac1 and <i>RLTR45</i> retrotransposon chromatin, in contrast <i>IAP</i> LTR sequences are depleted.</p

Genome-wide retrotransposon targets of transcriptional repression mechanisms in mouse ES cells.

<p>(A, C, E) Histograms showing repeat probes that change expression at least 2 fold (p<0.01) in Dnmt TKO, <i>Eset^shRNA</i>, and <i>Hdac1^−/−</i> ES cells respectively. (B, D, F) Plots showing the behaviour of the selected retrotransposon probe populations in Dnmt TKO, <i>Eset^shRNA</i>, and <i>Hdac1^−/−</i> ES cells respectively. Retrotransposons are colour-coded according to the legend. Vertical lines indicate the 2 fold change cut-off used in panels A, C and E. Note the divergent behaviour of <i>IAP</i> and <i>RLTR45</i> retrotransposons in <i>Hdac1^−/−</i> ES cells in contrast to Dnmt TKO and <i>Eset^shRNA</i> ES cells.</p

Genome-wide repetitive element expression in <i>Tex19.1^−/−</i> testes.

<p>(A–C) MA-plots showing the mean expression level for each expressed probe in the <i>Tex19.1</i> testis Illumina Beadarray data plotted against the fold upregulation of that probe in <i>Tex19.1^−/−</i> testes. Probes for repeat families (A), classes of LTR retrotransposons (B), and the <i>MMERVK10C</i> element (C) are colour-coded in each plot according to the legend. Note the group of six <i>MMERVK10C</i> ERVK LTR retrotransposon probes upregulated in <i>Tex19.1^−/−</i> testes. (D) Plot showing the behaviour of the entire <i>MMERVK10C</i> probe population in <i>Tex19.1^−/−</i> testes. Vertical lines indicate a 2 fold change. (E) qRT-PCR verification of <i>MMERVK10C</i> upregulation in C57BL/6 <i>Tex19.1^−/−</i> testes. Expression levels for each repetitive element (mean ± standard error for three animals) were normalized to <i>β-Actin</i> and expressed relative to littermate controls. Representative LTR retrotransposons belonging to ERV1, ERVK and ERVL classes do not change expression in <i>Tex19.1^−/−</i> testes. <i>Sdmg1</i> is a single-copy control gene for Sertoli cell expression to verify normalization between animals. MMERVK10C env.c and LINE1 ORF2 primer sets (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486.s007" target="_blank">Figure S2</a>) were used to assess <i>MMERVK10C</i> and <i>LINE-1</i> expression. Asterisk indicates a statistically significant difference (p<0.05).</p

Number of different repetitive elements represented by complementary probes in mouse gene expression microarray platforms.

<p>Mouse genome data is derived from Repeatmasker annotation of the mm9 assembly of the sequenced genome downloaded from the UCSC genome browser <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486-Fujita1" target="_blank">[62]</a>. The number of elements within each repetitive element class that are represented in the mouse genome and in the different microarray platforms is indicated. The number of genomic loci or microarray probes corresponding to each repetitive element class is also shown.</p

Differential regulation of retrotransposon genomic loci.

<p>(A) Plot showing the differential behaviour of different <i>RLTR4</i> retrotransposon probe populations in <i>Ring1B^−/−</i> single knockou<i>t ES</i> cells. Different <i>RLTR4</i> probe populations are colour-coded as shown in the legend, and vertical lines indicate a 4 fold change. (B) qRT-PCR verification of repetitive element expression in <i>Ring1B^−/−</i> ES cells. Expression levels (mean ± standard error) were normalized to <i>β-Actin</i> and expressed relative to wild-type control ES cells. MMERVK10C env.c and LINE1 5′UTR primer sets (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486.s007" target="_blank">Figure S2</a>) were used to assess <i>MMERVK10C</i> and <i>LINE-1</i> expression. The asterisk indicates a statistically significant difference (p<0.05). Note that different primers for <i>RLTR4</i> elements behave differently in the qRT-PCR assay. (C) qRT-PCR for different <i>MMERVK10C</i> primer sets in <i>Tex19.1^−/−</i> knockout and littermate control testes at 16 dpp. Expression levels (mean ± standard error for three animals) were normalized to <i>β-Actin</i> and expressed relative to littermate controls. Asterisks indicate statistically significant differences (p<0.05) (D) Plot showing the <i>MMERVK10C</i> genomic contigs flanked by <i>RLTR10C</i> LTRs that match only upregulated probes (blue), only unaffected probes (brown), neither class of probes (grey), or both classes of probe (green) in <i>Tex19.1^−/−</i> testes. Each contig is represented by a horizontal line that indicates the regions of the <i>MMERVK10C</i> sequence within it. The upregulated <i>MMERVK10C</i> contigs appear to contain recurrent deletions and may be non-autonomous. The positions of the qRT-PCR primers used in (C) are shaded orange. (E) Plot showing the bimodal behaviour of <i>IAP-int</i> retrotransposon probe populations in Dnmt TKO ES cells. Vertical lines indicate a 4 fold change. (F) qRT-PCR for of repetitive elements in Dnmt TKO ES cells. Expression levels (mean ± standard error) were normalized to <i>Gapdh</i> and expressed relative to wild-type control ES cells. The asterisk indicates a statistically significant difference (p<0.05). The LINE1 5′UTR.b primer set (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002486#pcbi.1002486.s007" target="_blank">Figure S2</a>) was used to assess <i>LINE-1</i> expression. Note the difference in behaviour between the two IAP-int primer sets. The <i>IAP</i> contig carrying deletions in the AP-1 binding site shown in panel G (IAP_chr10 primers) is expressed but not upregulated in Dnmt TKO ES cells. (G) Sequence alignment between an LTR of a full-length <i>IAP</i> element that does not change expression in Dnmt TKO ES cells (IAP_chr10), and the consensus sequence for the LTR (IAPLTR1a_Mm). The 10 bp deletion removes the AP-1 transcription factor binding site in the LTR.</p

Number of probes matching repetitive elements in mouse gene expression microarray platforms.

<p>Number of probes matching repetitive elements in mouse gene expression microarray platforms.</p